text: 8  (cid:115) (cid:115)  outage for a generic active UE in the jth tier is then given by  (cid:88)   the following theorem.   O − − K λ θ ρ(k) θ ρ(k) (j) = 1 exp k j o arctan j o s Λ ρ(j) ρ(j) Theorem 3: In a K-tier Poisson cellular network with a k=1 o o (18) common path-loss exponent η and truncated channel inversion (cid:26) (cid:114) (cid:18)(cid:114) (cid:19)(cid:27) (cid:80) power control where each tier has BS intensity λk and cutoff (cid:90) tUhEresshinoledacρh(okt)i,erincotnhsetituuptelinank,inadsseupmenidnegntthPaPtPthaendinttheartfetrhienigr ∞ exp − K λk xρ(ok) arctan xρ(ok) transmit powers are independent, the SINR outage probability R = k=1 Λ ρ(oj) ρ(oj) dx. for a generic active UE in the jt(cid:32)h tier is g(cid:33)iven by j 0 x + 1 (19)  (cid:88)  2 Equations (18) and (19) show that, in general, the SINR O(j) = 1 − exp −θjσ2 − K θj(ρ(ok)) η  outage probability and spectral efﬁciency in a certain tier s (cid:18) (cid:16)ρ(oj)(cid:17) (cid:19)k=(cid:90)1 ρ(oj)  dinetpeennsditieosn. tThheereSlIaNtivRe ocuuttaogffe tphrroesbhaoblidlistyanddectrheeasreeslaatinvde BthSe spectral efﬁciency improves as the cutoff threshold in the target  2  Λ2λkγ1 −2e,−ππΛΛ(cid:32)ρρPP(o(okuku))(cid:33)η2η  (cid:32)∞θjρρ(oj(o)k) (cid:33)−η1 yη y+ 1dy . (16) ttasihieghernigBianhlScerprienoacwtsueetenosrsfiaftanitnedhsdrteihnlseohwtcohuleedtrooifcntfhuttethhorrefetfisteahtrrhosgrlededtssehtciiorneelrdathissneec.aornTetdhah/esiorserstiiseltohrbwseeaecunrasdueB/fsoueSrl, intensities in the other tiers reduce the cross-tier interference. Theorems 3 and 4 show an important difference between multi-tier and single-tier cellular networks. That is, regardless of the maximum transmit power value P (i.e., binding or Proof See Appendix . u non-binding maximum transmit power constraint), the SINR outage probability and spectral efﬁciency in multi-tier cellular The SINR outage in (16) reduces to the single-tier case networks depend on the relative values of the cutoff thresholds given in (5), but with intensity Λ, when all tiers have the and the relative BS intensities. However, note that if all same cutoff threshold ρo despite of their different intensities. the tiers have the same cutoff threshold, regardless of the The spectral efﬁciency in the multi-tier case can be obtained relative BSs intensities, for an interference-limited scenario from the following theorem. with η = 4 and inﬁnite P , (18) and (19) reduce to (11) u and (12), respectively. Hence, the multi-tier cellular network Theorem 4: In a K-tier Poisson cellular network with a can be reduced to a single-tier network with intensity Λ, and common path-loss exponent η and truncated channel inversion both the SINR outage probability and the average spectral power control where each tier has BS intensity λ and cutoff efﬁciency become independent of the BS intensities in the k tihnreeaschholtdieρr(ocko)n, sintittuhteeuapilnindkep, eanssduemntinPgPPthaantdthtehaintttehrefierrtirnagnsUmEist different tiers if Pu is non-binding. It is worth mentioning that similar performance was observed in [2] for the downlink powers are independent, the average spectral efﬁciency of multi-tier case. In particular, it was shown that if each UE transmission by an active UE in the jth tier is given by (cid:90)  (cid:88) (cid:32) (cid:33) associates to the BS offering the highest average received power, the outage probability and spectral efﬁciency become Rj = 0∞(cid:18)x +1 1 (cid:16)exp (cid:17)−xρ(cid:19)(oσj2)(cid:90)− kK=1 xρρ(o(ojk)) η2  iBnSdsepiennddiefnfetroenf tthneetrweloartkivteieirnst.ensities and transmit powers of  2  B. Different Path-loss Exponents Λ2λkγ1 −2e,−ππΛΛ(cid:32)ρρPP(o(okuku))(cid:33)η2η  (cid:32)∞xρρ(o(ojk)) (cid:33)−η1 yη y+ 1dy dx. (17) wofoItrnhkethtUieeErpssresdvhoiaeorsuesntohsteeccsthiaoamnng,eeitpfahrtaohsm-lboteshesantesixnhpootwhneennstithn,agtthlewe-thaieesnsrocacelillaltnuieloatn-r networks. That is, the association regions of the BSs will form a Voronoi tessellation. On the other hand, if different tiers have different path-loss exponents, the association of UEs in the multi-tier case deviates from that in the single-tier case where Proof The proof is similar to the one in Appendix . the association regions of the BSs form a weighted Voronoi tessellation. That is, the BSs in tiers with lower path-loss For brevity, we consider only one special case. For inﬁnite exponents will have larger service areas than BSs in tiers with P , interference-limited network scenario, and η = 4, the higher path-loss exponents. The transmit power in a multi- u SINR outage and average spectral efﬁciency in the uplink in tier cellular network with different path-loss exponents can be the jth tier reduce to the following: characterized by the following lemma. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: outage

text: for

text: a

text: generic

text: active

text: UE

text: in

text: the

text: jth

text: tier

text: is

text: then

text: given

text: by

text: the

text: following

text: theorem.

text: O

text: K

text: λ

text: θ

text: ρ(k)

text: θ

text: ρ(k)

text: (j)

text: exp

text: k

text: j

text: o

text: arctan

text: j

text: o

text: s

text: Λ

text: ρ(j)

text: ρ(j)

text: Theorem

text: In

text: a

text: K-tier

text: Poisson

text: cellular

text: network

text: with

text: a

text: o

text: o

text: common

text: path-loss

text: exponent

text: η

text: and

text: truncated

text: channel

text: inversion

text: power

text: control

text: where

text: each

text: tier

text: has

text: BS

text: intensity

text: λk

text: and

text: cutoff

text: exp

text: K

text: λk

text: xρ(ok)

text: arctan

text: xρ(ok)

text: transmit

text: powers

text: are

text: independent,

text: the

text: SINR

text: outage

text: probability

text: R

text: Λ

text: ρ(oj)

text: ρ(oj)

text: dx.

text: for

text: a

text: generic

text: active

text: UE

text: in

text: the

text: tier

text: is

text: by

text: j

text: x

text: Equations

text: and

text: show

text: that,

text: in

text: general,

text: the

text: SINR

text: O(j)

text: exp

text: −θjσ2

text: K

text: θj(ρ(ok))

text: η

text: outage

text: probability

text: and

text: spectral

text: efﬁciency

text: in

text: a

text: certain

text: tier

text: s

text: ρ(oj)

text: dinetpeennsditieosn.

text: tThheereSlIaNtivRe

text: ocuuttaogffe

text: BthSe

text: spectral

text: efﬁciency

text: improves

text: as

text: the

text: cutoff

text: threshold

text: in

text: the

text: target

text: Λ2λkγ1

text: yη

text: y+

text: 1dy

text: intensities

text: in

text: the

text: other

text: tiers

text: reduce

text: the

text: cross-tier

text: interference.

text: Theorems

text: and

text: show

text: an

text: important

text: difference

text: between

text: multi-tier

text: and

text: single-tier

text: cellular

text: networks.

text: That

text: is,

text: regardless

text: of

text: the

text: maximum

text: transmit

text: power

text: value

text: P

text: (i.e.,

text: binding

text: or

text: Proof

text: See

text: Appendix

text: u

text: non-binding

text: maximum

text: transmit

text: power

text: constraint),

text: the

text: SINR

text: outage

text: probability

text: and

text: spectral

text: efﬁciency

text: in

text: multi-tier

text: cellular

text: The

text: SINR

text: outage

text: in

text: reduces

text: to

text: the

text: single-tier

text: case

text: networks

text: depend

text: on

text: the

text: relative

text: values

text: of

text: the

text: cutoff

text: thresholds

text: given

text: in

text: but

text: with

text: intensity

text: Λ,

text: when

text: all

text: tiers

text: have

text: the

text: and

text: the

text: relative

text: BS

text: intensities.

text: However,

text: note

text: that

text: if

text: all

text: same

text: cutoff

text: threshold

text: ρo

text: despite

text: of

text: their

text: different

text: intensities.

text: the

text: tiers

text: have

text: the

text: same

text: cutoff

text: threshold,

text: regardless

text: of

text: the

text: The

text: spectral

text: efﬁciency

text: in

text: the

text: multi-tier

text: case

text: can

text: be

text: obtained

text: relative

text: BSs

text: intensities,

text: for

text: an

text: interference-limited

text: scenario

text: from

text: the

text: following

text: theorem.

text: with

text: η

text: and

text: inﬁnite

text: P

text: and

text: reduce

text: to

text: u

text: and

text: respectively.

text: Hence,

text: the

text: multi-tier

text: cellular

text: network

text: Theorem

text: In

text: a

text: K-tier

text: Poisson

text: cellular

text: network

text: with

text: a

text: can

text: be

text: reduced

text: to

text: a

text: single-tier

text: network

text: with

text: intensity

text: Λ,

text: and

text: common

text: path-loss

text: exponent

text: η

text: and

text: truncated

text: channel

text: inversion

text: both

text: the

text: SINR

text: outage

text: probability

text: and

text: the

text: average

text: spectral

text: power

text: control

text: where

text: each

text: tier

text: has

text: BS

text: intensity

text: λ

text: and

text: cutoff

text: efﬁciency

text: become

text: independent

text: of

text: the

text: BS

text: intensities

text: in

text: the

text: k

text: tihnreeaschholtdieρr(ocko)n,

text: sintittuhteeuapilnindkep,

text: different

text: tiers

text: if

text: Pu

text: is

text: non-binding.

text: It

text: is

text: worth

text: mentioning

text: that

text: similar

text: performance

text: was

text: observed

text: in

text: for

text: the

text: downlink

text: powers

text: are

text: independent,

text: the

text: average

text: spectral

text: efﬁciency

text: of

text: multi-tier

text: case.

text: In

text: particular,

text: it

text: was

text: shown

text: that

text: if

text: each

text: UE

text: transmission

text: by

text: an

text: active

text: UE

text: in

text: the

text: jth

text: tier

text: is

text: given

text: by

text: associates

text: to

text: the

text: BS

text: offering

text: the

text: highest

text: average

text: received

text: power,

text: the

text: outage

text: probability

text: and

text: spectral

text: efﬁciency

text: become

text: Rj

text: xρρ(o(ojk))

text: η2

text: iBnSdsepiennddiefnfetroenf

text: tthneetrweloartkivteieirnst.ensities

text: and

text: transmit

text: powers

text: of

text: B.

text: Different

text: Path-loss

text: Exponents

text: Λ2λkγ1

text: yη

text: y+

text: 1dy

text: dx.

text: networks.

text: That

text: is,

text: the

text: association

text: regions

text: of

text: the

text: BSs

text: will

text: form

text: a

text: Voronoi

text: tessellation.

text: On

text: the

text: other

text: hand,

text: if

text: different

text: tiers

text: have

text: different

text: path-loss

text: exponents,

text: the

text: association

text: of

text: UEs

text: in

text: the

text: multi-tier

text: case

text: deviates

text: from

text: that

text: in

text: the

text: single-tier

text: case

text: where

text: Proof

text: The

text: proof

text: is

text: similar

text: to

text: the

text: one

text: in

text: Appendix

text: the

text: association

text: regions

text: of

text: the

text: BSs

text: form

text: a

text: weighted

text: Voronoi

text: tessellation.

text: That

text: is,

text: the

text: BSs

text: in

text: tiers

text: with

text: lower

text: path-loss

text: For

text: brevity,

text: we

text: consider

text: only

text: one

text: special

text: case.

text: For

text: inﬁnite

text: exponents

text: will

text: have

text: larger

text: service

text: areas

text: than

text: BSs

text: in

text: tiers

text: with

text: P

text: interference-limited

text: network

text: scenario,

text: and

text: η

text: the

text: higher

text: path-loss

text: exponents.

text: The

text: transmit

text: power

text: in

text: a

text: multi-

text: u

text: SINR

text: outage

text: and

text: average

text: spectral

text: efﬁciency

text: in

text: the

text: uplink

text: in

text: tier

text: cellular

text: network

text: with

text: different

text: path-loss

text: exponents

text: can

text: be

text: the

text: jth

text: tier

text: reduce

text: to

text: the

text: following:

text: characterized

text: by

text: the

text: following

text: lemma.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##


text: Bridgeland Stability of Line Bundles on Surfaces Daniele Arcara and Eric Miles Abstract We study the Bridgeland stability of line bundles on surfaces with respect to certain Bridgeland stability conditions determined by divi- sors. Given a smooth projective surface S, we show that a line bundle L is always Bridgeland stable for those stability conditions if there ⊆ are no curves C S of negative self-intersection. When a curve C − of negative self-intersection is present, L is destabilized by L( C) for some stability conditions. We conjecture that line bundles of the form − L( C) are the only objects that can destabilize L and that torsion | sheaves of the form L(C) are the only objects that can destabi- C lize L[1]. We prove our conjecture in several cases, in particular for Hirzebruch surfaces. Contents 1 Introduction 1 2 Bridgeland stability conditions 3 2.1 General deﬁnition of Bridgeland stability conditions . . . . . . 3 2.2 Bridgeland stability conditions on a surface . . . . . . . . . . . 5 2.3 Slices of Stab (S) . . . . . . . . . . . . . . . . . . . . . . . . 6 div O 3 Reduction to the case of 7 S O 4 Preliminaries on the stability of 8 O S 4.1 Subobjects of and their walls . . . . . . . . . . . . . . . . 8 S O 4.2 Bridgeland stability of for t >> 0 . . . . . . . . . . . . . . 13 S 1

text: Bridgeland

label: title

text: Stability

text: of

text: Line

text: Bundles

text: on

text: Surfaces

text: Daniele

label: author

text: Arcara

text: and

text: Eric

text: Miles

text: Abstract

label: paragraph

text: We

label: abstract

text: study

text: the

text: Bridgeland

text: stability

text: of

text: line

text: bundles

text: on

text: surfaces

text: with

text: respect

text: to

text: certain

text: Bridgeland

text: stability

text: conditions

text: determined

text: by

text: divi-

text: sors.

text: Given

text: a

text: smooth

text: projective

text: surface

text: S,

text: we

text: show

text: that

text: a

text: line

text: bundle

text: L

text: is

text: always

text: Bridgeland

text: stable

text: for

text: those

text: stability

text: conditions

text: if

text: there

text: are

text: no

text: curves

text: C

text: S

text: of

text: negative

text: self-intersection.

text: When

text: a

text: curve

text: C

text: of

text: negative

text: self-intersection

text: is

text: present,

text: L

text: is

text: destabilized

text: by

text: L(

text: C)

text: for

text: some

text: stability

text: conditions.

text: We

text: conjecture

text: that

text: line

text: bundles

text: of

text: the

text: form

text: L(

text: C)

text: are

text: the

text: only

text: objects

text: that

text: can

text: destabilize

text: L

text: and

text: that

text: torsion

text: sheaves

text: of

text: the

text: form

text: L(C)

text: are

text: the

text: only

text: objects

text: that

text: can

text: destabi-

text: C

text: lize

text: We

text: prove

text: our

text: conjecture

text: in

text: several

text: cases,

text: in

text: particular

text: for

text: Hirzebruch

text: surfaces.

text: Contents

text: Introduction

text: Bridgeland

text: stability

text: conditions

text: General

text: deﬁnition

text: of

text: Bridgeland

text: stability

text: conditions

text: Bridgeland

text: stability

text: conditions

text: on

text: a

text: surface

text: Slices

text: of

text: Stab

text: (S)

text: div

text: O

text: Reduction

text: to

text: the

text: case

text: of

text: S

text: O

text: Preliminaries

text: on

text: the

text: stability

text: of

text: O

text: S

text: Subobjects

text: of

text: and

text: their

text: walls

text: S

text: O

text: Bridgeland

text: stability

text: of

text: for

text: t

text: S


text: Below is a summary of all possible weakly destabilizing walls (pictures drawn for d < 0 and d > 0): h g − Parabola. When d2 d2 = 0 and c > 0. g h − Ellipse. When d2 d2 < 0 and c > 0. g h − Cone. When d2 d2 > 0 and c = 0. g h − Right Hyperbola. When d2 d2 > 0 and c > 0. g h − Left Hyperbola. When d2 d2 > 0 and c < 0. g h Figure 2: Parabola Figure 3: Ellipse Figure 4: Cone Figure 5: Right hyperbola Figure 6: Left hyperbola We end this section by noting an important geometric property of these walls that will be useful in various proofs later in the paper. O A Lemma 4.3. Given two subobjects E and E of in , there exists at 1 2 S most one non-zero value u of u such that 0 W O ∩ W O ∩ (E , ) Π = (E , ) Π , 1 S u0 2 S u0 unless the two walls coincide everywhere. 13 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: Below

text: is

text: a

text: summary

text: of

text: all

text: possible

text: weakly

text: destabilizing

text: walls

text: (pictures

text: drawn

text: for

text: d

text: and

text: d

text: h

text: g

text: Parabola.

text: When

text: and

text: c

text: g

text: h

text: Ellipse.

text: When

text: and

text: c

text: g

text: h

text: Cone.

text: When

text: and

text: c

text: g

text: h

text: Right

text: Hyperbola.

text: When

text: and

text: c

text: g

text: h

text: Left

text: Hyperbola.

text: When

text: and

text: c

text: g

text: h

text: Figure

text: Parabola

label: caption

text: Figure

text: Ellipse

text: Figure

text: Cone

text: Figure

text: Right

text: hyperbola

text: Figure

text: Left

text: hyperbola

text: We

text: end

text: this

text: section

text: by

text: noting

text: an

text: important

text: geometric

text: property

text: of

text: these

text: walls

text: that

text: will

text: be

text: useful

text: in

text: various

text: proofs

text: later

text: in

text: the

text: paper.

text: O

text: A

text: Lemma

text: Given

text: two

text: subobjects

text: E

text: and

text: E

text: of

text: in

text: there

text: exists

text: at

text: S

text: most

text: one

text: non-zero

text: value

text: u

text: of

text: u

text: such

text: that

text: W

text: O

text: W

text: O

text: (E

text: Π

text: (E

text: Π

text: S

text: S

text: unless

text: the

text: two

text: walls

text: coincide

text: everywhere.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: ⊆ ⊆ ⊆ · · · ⊆ ⊆ Let 0 = E E E E − E = E be the Harder- 0 1 2 n 1 n Narasimhan ﬁltration of E, and let K = E /E − (so that K = E and i i i 1 · · · 1 1 K = E/E − ). We have that µ (K ) > µ (K ) > > µ (K ). In n n 1 H 1 H 2 H n addition, let K = K . n ⊆ ⊆ ⊆ · · · ⊆ ⊆ − Similarly, let 0 = F F F F − F = H 1(Q) be the 0 1 −2 m 1 m Harder-Narasimhan ﬁltration of H 1(Q), and let J = F /F − with J = J . ⊆ O ∈ A ≤i i i 1 1 Remark 4.6. Note that E iﬀ µ (J) s < µ (K). Moreover, S H H we must have that µ (J) < µ (K) < 0. H H O A Proof. For E to be a subobject of in , we need in particular that − A S A both E and H 1(Q) be objects of . For E to be an object of , we must − A have that s < µ (K), and for H 1(Q) to be an object of , we must have ≥ H that s µ (J). The last condition that µ (K) < 0 originates from the H H I − − fact that, as we observed above, E is an extension of ( C) by H 1(Q) Z for some eﬀective curve C and zero-dimensional scheme Z (with C or Z O ∈ A possibly zero). Because implies that s < 0 by Remark 2.3, then S − ≤ µ (J) < s < 0 implies that µ (H 1(Q)) µ (J) < 0. Because E is an H I − − H H extension of ( C) by H 1(Q), we must also have that µ (E) < 0, and Z ≤ H therefore, µ (K) µ (E) < 0. H H The idea for Bertram’s lemma is the following: For every value of u, the W O ∩ wall (E, ) Π is a semicircle in the upper-half plane Π parametrized by S u u (s, t). The lemma shows that if the semicircle intersects the lines s = µ (J) (cid:48) ⊆ O H or s = µ (K), then there exists another subobject E such that W O H W (cid:48) O W O ∩ S (E, ) is inside (E , ) at u. If the semicircle (E, ) Π intersects S S(cid:48) S u the line s = µ (K), then E = E − , and if it intersects the line s = µ (J), (cid:48) H n 1 ⊆ − ⊆ H then E = E/J (note that, as sheaves, J = J = F H 1(Q) E). 1 1 ⊆ O Lemma 4.7 (Bertram’s Lemma). Fix H and G as above, and let E A Z A ∈ W O S in for some σ = σ = ( , ) such that σ (E, ). 0 0 s0H+u0G,t0H 0 0 S W O ∩ (1) If (E, ) Π intersects the line s = µ (K) for t > 0, then S u0 ⊆ O A H β(E − ) > β(E) at σ , with E − in (in particular, E is not n 1 0 n 1 S 0 µ -semistable). H W O ∩ (2) If (E, ) Π intersects the line s = µ (J) for t > 0, then S u0 ⊆ O A H β(E/J) > β(E) at σ , with E/J in . 0 S 0 W O ∩ Proof. (1) Maciocia proved in [7] that if E is µ -semistable, then (E, ) H W O ∩S Π does not intersect the line s = µ (E) for any u. Therefore, if (E, ) u H ∈ SA Π intersects s = µ (K), E cannot be µ -semistable. Because E , u0 H H 0 15 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Let

text: E

text: E

text: E

text: E

text: E

text: E

text: be

text: the

text: Harder-

text: n

text: n

text: Narasimhan

text: ﬁltration

text: of

text: E,

text: and

text: let

text: K

text: E

text: /E

text: (so

text: that

text: K

text: E

text: and

text: i

text: i

text: i

text: K

text: E/E

text: We

text: have

text: that

text: µ

text: (K

text: µ

text: (K

text: µ

text: (K

text: In

text: n

text: n

text: H

text: H

text: H

text: n

text: addition,

text: let

text: K

text: K

text: n

text: Similarly,

text: let

text: F

text: F

text: F

text: F

text: F

text: H

text: be

text: the

text: −2

text: m

text: m

text: Harder-Narasimhan

text: ﬁltration

text: of

text: H

text: and

text: let

text: J

text: F

text: /F

text: with

text: J

text: J

text: O

text: A

text: ≤i

text: i

text: i

text: Remark

text: Note

text: that

text: E

text: iﬀ

text: µ

text: (J)

text: s

text: µ

text: (K).

text: Moreover,

text: S

text: H

text: H

text: we

text: must

text: have

text: that

text: µ

text: (J)

text: µ

text: (K)

text: H

text: H

text: O

text: A

text: Proof.

text: For

text: E

text: to

text: be

text: a

text: subobject

text: of

text: in

text: we

text: need

text: in

text: particular

text: that

text: A

text: S

text: A

text: both

text: E

text: and

text: H

text: be

text: objects

text: of

text: For

text: E

text: to

text: be

text: an

text: object

text: of

text: we

text: must

text: A

text: have

text: that

text: s

text: µ

text: (K),

text: and

text: for

text: H

text: to

text: be

text: an

text: object

text: of

text: we

text: must

text: have

text: H

text: that

text: s

text: µ

text: (J).

text: The

text: last

text: condition

text: that

text: µ

text: (K)

text: originates

text: from

text: the

text: H

text: H

text: I

text: fact

text: that,

text: as

text: we

text: observed

text: above,

text: E

text: is

text: an

text: extension

text: of

text: C)

text: by

text: H

text: Z

text: for

text: some

text: eﬀective

text: curve

text: C

text: and

text: zero-dimensional

text: scheme

text: Z

text: (with

text: C

text: or

text: Z

text: O

text: A

text: possibly

text: zero).

text: Because

text: implies

text: that

text: s

text: by

text: Remark

text: then

text: S

text: µ

text: (J)

text: s

text: implies

text: that

text: µ

text: (H

text: µ

text: (J)

text: Because

text: E

text: is

text: an

text: H

text: I

text: H

text: H

text: extension

text: of

text: C)

text: by

text: H

text: we

text: must

text: also

text: have

text: that

text: µ

text: (E)

text: and

text: Z

text: H

text: therefore,

text: µ

text: (K)

text: µ

text: (E)

text: H

text: H

text: The

text: idea

text: for

text: Bertram’s

text: lemma

text: is

text: the

text: following:

text: For

text: every

text: value

text: of

text: u,

text: the

text: W

text: O

text: wall

text: (E,

text: Π

text: is

text: a

text: semicircle

text: in

text: the

text: upper-half

text: plane

text: Π

text: parametrized

text: by

text: S

text: u

text: u

text: (s,

text: t).

text: The

text: lemma

text: shows

text: that

text: if

text: the

text: semicircle

text: intersects

text: the

text: lines

text: s

text: µ

text: (J)

text: O

text: H

text: or

text: s

text: µ

text: (K),

text: then

text: there

text: exists

text: another

text: subobject

text: E

text: such

text: that

text: W

text: O

text: H

text: W

text: O

text: W

text: O

text: S

text: (E,

text: is

text: inside

text: (E

text: at

text: u.

text: If

text: the

text: semicircle

text: (E,

text: Π

text: intersects

text: S

text: S

text: u

text: the

text: line

text: s

text: µ

text: (K),

text: then

text: E

text: E

text: and

text: if

text: it

text: intersects

text: the

text: line

text: s

text: µ

text: (J),

text: H

text: n

text: H

text: then

text: E

text: E/J

text: (note

text: that,

text: as

text: sheaves,

text: J

text: J

text: F

text: H

text: E).

text: O

text: Lemma

text: (Bertram’s

text: Lemma).

text: Fix

text: H

text: and

text: G

text: as

text: above,

text: and

text: let

text: E

text: A

text: Z

text: A

text: W

text: O

text: S

text: in

text: for

text: some

text: σ

text: σ

text: such

text: that

text: σ

text: (E,

text: S

text: W

text: O

text: If

text: (E,

text: Π

text: intersects

text: the

text: line

text: s

text: µ

text: (K)

text: for

text: t

text: then

text: S

text: O

text: A

text: H

text: β(E

text: β(E)

text: at

text: σ

text: with

text: E

text: in

text: (in

text: particular,

text: E

text: is

text: not

text: n

text: n

text: S

text: µ

text: -semistable).

text: H

text: W

text: O

text: If

text: (E,

text: Π

text: intersects

text: the

text: line

text: s

text: µ

text: (J)

text: for

text: t

text: then

text: S

text: O

text: A

text: H

text: β(E/J)

text: β(E)

text: at

text: σ

text: with

text: E/J

text: in

text: S

text: W

text: O

text: Proof.

text: Maciocia

text: proved

text: in

text: that

text: if

text: E

text: is

text: µ

text: -semistable,

text: then

text: (E,

text: H

text: W

text: O

text: ∩S

text: Π

text: does

text: not

text: intersect

text: the

text: line

text: s

text: µ

text: (E)

text: for

text: any

text: u.

text: Therefore,

text: if

text: (E,

text: u

text: H

text: SA

text: Π

text: intersects

text: s

text: µ

text: (K),

text: E

text: cannot

text: be

text: µ

text: -semistable.

text: Because

text: E

text: H

text: H

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: O 5 Bridgeland stability of S In this section, we prove the ﬁrst part of Conjecture 1 for surfaces with any Picard rank and no curves of negative self-intersection as well as surfaces with Picard rank 2 and one irreducible curve of negative self-intersection. Proposition 5.10 serves as evidence for the conjecture on surfaces with Picard rank 2 and two irreducible curves of negative self-intersection. O 5.1 Subobjects of of rank 1 S O Understanding the rank-1 weakly destabilizing subobjects of is crucial ⊆S O to our main results, all of which use induction. If a subobject E is of I − S rank 1, then Lemma 4.1 shows that E must be equal to ( C) for some Z eﬀective curve C and some zero-dimensional scheme Z, with C or Z possibly 0. I If C = 0, then E = for some zero-dimensional scheme Z. In this case, I OZ I − − − − does not destabilize because β( ) = ( 2l(Z) + s2 u2 t2)/( 2st), Z S O Z − − − where l(Z) is the length of Z, and β( ) = (s2 u2 t2)/( 2st). Therefore, O ∈ A S I O I when , we have that s < 0, and β( ) < β( ). This means that S O O ∈ A Z S Z does not destabilize whenever . S (cid:54) S I − ⊆ O Assume now that C = 0, and consider the subobject ( C) , Z S with Z a zero-dimensional scheme Z (possibly equal to 0). We show that for I − O ( C) to weakly destabilize , we must have C2 < 0. Z ≥ S I − O Proposition 5.1. If C2 0, then ( C) does not weakly destabilize Z I − ⊆ O A S for any σ, i.e., there does not exist any σ such that ( C) in and I − O Z S β( ( C)) = β( ). Z S I − − − Proof. Note that ch( ( C)) = (1, C, C2/2 l(Z)), where l(Z) is the Z length of Z. If C = c H + c G + α , with α .H = α .G = 0, then C2 = − ≤h g C C C ≥ c2 c2 +α2 , and α2 0 by the Hodge Index Theorem. Therefore, if C2 0, h g − C ≥ C W I − O then c2 c2 0. The equation for the wall ( ( C), ) simpliﬁes to h g Z S − − − c (s2 + t2 + u2) 2c su + (c2 c2)s + α2 s 2l(Z)s = 0. h g h g C Note that because C is eﬀective and because H is ample, c = C.H > 0. − h If c2 c2 > 0, then the wall at t = 01 is an ellipse going through (0, 0) h g 1It is suﬃcient to consider the walls at t = 0 for the following reason: Because the O W O ⊂ S walls for in Π are semicircles [7], the wall (E, ) lies inside the vertical S W u O S G,H cylinder over (E, ) at t = 0. S 17 ##LTLine##

text: O

label: section

text: Bridgeland

text: stability

text: of

text: S

text: In

text: this

text: section,

text: we

text: prove

text: the

text: ﬁrst

text: part

text: of

text: Conjecture

text: for

text: surfaces

text: with

text: any

text: Picard

text: rank

text: and

text: no

text: curves

text: of

text: negative

text: self-intersection

text: as

text: well

text: as

text: surfaces

text: with

text: Picard

text: rank

text: and

text: one

text: irreducible

text: curve

text: of

text: negative

text: self-intersection.

text: Proposition

text: serves

text: as

text: evidence

text: for

text: the

text: conjecture

text: on

text: surfaces

text: with

text: Picard

text: rank

text: and

text: two

text: irreducible

text: curves

text: of

text: negative

text: self-intersection.

text: O

text: Subobjects

text: of

text: of

text: rank

text: S

text: O

text: Understanding

text: the

text: weakly

text: destabilizing

text: subobjects

text: of

text: is

text: crucial

text: ⊆S

text: O

text: to

text: our

text: main

text: results,

text: all

text: of

text: which

text: use

text: induction.

text: If

text: a

text: subobject

text: E

text: is

text: of

text: I

text: S

text: rank

text: then

text: Lemma

text: shows

text: that

text: E

text: must

text: be

text: equal

text: to

text: C)

text: for

text: some

text: Z

text: eﬀective

text: curve

text: C

text: and

text: some

text: zero-dimensional

text: scheme

text: Z,

text: with

text: C

text: or

text: Z

text: possibly

text: I

text: If

text: C

text: then

text: E

text: for

text: some

text: zero-dimensional

text: scheme

text: Z.

text: In

text: this

text: case,

text: I

text: OZ

text: I

text: does

text: not

text: destabilize

text: because

text: β(

text: Z

text: S

text: O

text: Z

text: where

text: l(Z)

text: is

text: the

text: length

text: of

text: Z,

text: and

text: β(

text: Therefore,

text: O

text: A

text: S

text: I

text: O

text: I

text: when

text: we

text: have

text: that

text: s

text: and

text: β(

text: β(

text: This

text: means

text: that

text: S

text: O

text: O

text: A

text: Z

text: S

text: Z

text: does

text: not

text: destabilize

text: whenever

text: S

text: S

text: I

text: O

text: Assume

text: now

text: that

text: C

text: and

text: consider

text: the

text: subobject

text: C)

text: Z

text: S

text: with

text: Z

text: a

text: zero-dimensional

text: scheme

text: Z

text: (possibly

text: equal

text: to

text: We

text: show

text: that

text: for

text: I

text: O

text: C)

text: to

text: weakly

text: destabilize

text: we

text: must

text: have

text: Z

text: S

text: I

text: O

text: Proposition

text: If

text: then

text: C)

text: does

text: not

text: weakly

text: destabilize

text: Z

text: I

text: O

text: A

text: S

text: for

text: any

text: σ,

text: i.e.,

text: there

text: does

text: not

text: exist

text: any

text: σ

text: such

text: that

text: C)

text: in

text: and

text: I

text: O

text: Z

text: S

text: β(

text: C))

text: β(

text: Z

text: S

text: I

text: Proof.

text: Note

text: that

text: ch(

text: C))

text: C,

text: l(Z)),

text: where

text: l(Z)

text: is

text: the

text: Z

text: length

text: of

text: Z.

text: If

text: C

text: c

text: H

text: c

text: G

text: α

text: with

text: α

text: .H

text: α

text: .G

text: then

text: ≤h

text: g

text: C

text: C

text: C

text: and

text: α2

text: by

text: the

text: Hodge

text: Index

text: Theorem.

text: Therefore,

text: if

text: h

text: g

text: C

text: C

text: W

text: I

text: O

text: then

text: The

text: equation

text: for

text: the

text: wall

text: C),

text: simpliﬁes

text: to

text: h

text: g

text: Z

text: S

text: c

text: su

text: α2

text: s

text: h

text: g

text: h

text: g

text: C

text: Note

text: that

text: because

text: C

text: is

text: eﬀective

text: and

text: because

text: H

text: is

text: ample,

text: c

text: C.H

text: h

text: If

text: then

text: the

text: wall

text: at

text: t

text: is

text: an

text: ellipse

text: going

text: through

text: h

text: g

text: is

text: suﬃcient

text: to

text: consider

text: the

text: walls

text: at

text: t

text: for

text: the

text: following

text: reason:

text: Because

text: the

text: O

text: W

text: O

text: S

text: walls

text: for

text: in

text: Π

text: are

text: semicircles

text: the

text: wall

text: (E,

text: lies

text: inside

text: the

text: vertical

text: S

text: W

text: u

text: O

text: S

text: G,H

text: cylinder

text: over

text: (E,

text: at

text: t

text: S

text: ##LTLine##


text: The goal of this paper is to prove the conjecture in several cases and provide evidence for the conjecture in others. Speciﬁcally, we prove the following. Theorem 1.1. The conjecture is true in the following cases: • If S does not have any curves of negative self-intersection. • If the Picard rank of S is 2 and there exists only one irreducible curve of negative self-intersection. In particular, the conjecture is true for Hirzebruch surfaces because they have Picard rank 2 and only one irreducible curve of negative self-intersection (see, e.g., [4]). We cite Propositions 5.5 and 5.10 as further evidence for our conjecture in general. Proposition 5.5 establishes some structure for the walls of an actually destabilizing subobject of a line bundle on surfaces of any Picard rank, and Proposition 5.10 proves a stronger version of the conjecture for a subset of stability conditions when S has Picard rank 2 and two irreducible curves of negative self-intersection. In [1], the stability of line bundles is proven for stability conditions σ D,H with D = sH and is utilized in [2] and [5] when classifying destabilizing walls (cid:54) for ideal sheaves of points on surfaces. When D = sH, the more algebraic proof of [1] using the Bogomolov inequality and Hodge Index Theorem fails, and new techniques are required. We use Theorem 3.1 (Bertram’s Nested Wall Theorem) from [7] and Lemma 6.3 (which we refer to as Bertram’s Lemma) from [2] along with an analysis of the relative geometry of relevant walls in certain three-dimensional slices of the space of stability conditions. We begin in Section by introducing Bridgeland stability conditions and the stability conditions σ of interest, as well as important slices of the D,H space of stability conditions. In Section , we present an action by line bundles O that allows us to consider only in our question concerning the stability S of line bundles. In Section , we consider the basic structure of subobjects of O and present two previously obtained results that will serve as important S tools in the remainder of this paper. In Section , we prove our main results, O but we ﬁrst consider the rank 1 subobjects of . The rank 1 subobjects S form the base case for our main results, all of which use induction. The case O of [1] is then primarily completed using duality, which allows us to use S O our results for except when D.H = 0. S 3

text: The

text: goal

text: of

text: this

text: paper

text: is

text: to

text: prove

text: the

text: conjecture

text: in

text: several

text: cases

text: and

text: provide

text: evidence

text: for

text: the

text: conjecture

text: in

text: others.

text: Speciﬁcally,

text: we

text: prove

text: the

text: following.

text: Theorem

text: The

text: conjecture

text: is

text: true

text: in

text: the

text: following

text: cases:

label: list

text: If

text: S

text: does

text: not

text: have

text: any

text: curves

text: of

text: negative

text: self-intersection.

text: If

text: the

text: Picard

text: rank

text: of

text: S

text: is

text: and

text: there

text: exists

text: only

text: one

text: irreducible

text: curve

text: of

text: negative

text: self-intersection.

text: In

text: particular,

text: the

text: conjecture

text: is

text: true

text: for

text: Hirzebruch

text: surfaces

text: because

text: they

text: have

text: Picard

text: rank

text: and

text: only

text: one

text: irreducible

text: curve

text: of

text: negative

text: self-intersection

text: (see,

text: e.g.,

text: We

text: cite

text: Propositions

text: and

text: as

text: further

text: evidence

text: for

text: our

text: conjecture

text: in

text: general.

text: Proposition

text: establishes

text: some

text: structure

text: for

text: the

text: walls

text: of

text: an

text: actually

text: destabilizing

text: subobject

text: of

text: a

text: line

text: bundle

text: on

text: surfaces

text: of

text: any

text: Picard

text: rank,

text: and

text: Proposition

text: proves

text: a

text: stronger

text: version

text: of

text: the

text: conjecture

text: for

text: a

text: subset

text: of

text: stability

text: conditions

text: when

text: S

text: has

text: Picard

text: rank

text: and

text: two

text: irreducible

text: curves

text: of

text: negative

text: self-intersection.

text: In

text: the

text: stability

text: of

text: line

text: bundles

text: is

text: proven

text: for

text: stability

text: conditions

text: σ

text: D,H

text: with

text: D

text: sH

text: and

text: is

text: utilized

text: in

text: and

text: when

text: classifying

text: destabilizing

text: walls

text: for

text: ideal

text: sheaves

text: of

text: points

text: on

text: surfaces.

text: When

text: D

text: sH,

text: the

text: more

text: algebraic

text: proof

text: of

text: using

text: the

text: Bogomolov

text: inequality

text: and

text: Hodge

text: Index

text: Theorem

text: fails,

text: and

text: new

text: techniques

text: are

text: required.

text: We

text: use

text: Theorem

text: (Bertram’s

text: Nested

text: Wall

text: Theorem)

text: from

text: and

text: Lemma

text: (which

text: we

text: refer

text: to

text: as

text: Bertram’s

text: Lemma)

text: from

text: along

text: with

text: an

text: analysis

text: of

text: the

text: relative

text: geometry

text: of

text: relevant

text: walls

text: in

text: certain

text: three-dimensional

text: slices

text: of

text: the

text: space

text: of

text: stability

text: conditions.

text: We

text: begin

text: in

text: Section

text: by

text: introducing

text: Bridgeland

text: stability

text: conditions

text: and

text: the

text: stability

text: conditions

text: σ

text: of

text: interest,

text: as

text: well

text: as

text: important

text: slices

text: of

text: the

text: D,H

text: space

text: of

text: stability

text: conditions.

text: In

text: Section

text: we

text: present

text: an

text: action

text: by

text: line

text: bundles

text: O

text: that

text: allows

text: us

text: to

text: consider

text: only

text: in

text: our

text: question

text: concerning

text: the

text: stability

text: S

text: of

text: line

text: bundles.

text: In

text: Section

text: we

text: consider

text: the

text: basic

text: structure

text: of

text: subobjects

text: of

text: O

text: and

text: present

text: two

text: previously

text: obtained

text: results

text: that

text: will

text: serve

text: as

text: important

text: S

text: tools

text: in

text: the

text: remainder

text: of

text: this

text: paper.

text: In

text: Section

text: we

text: prove

text: our

text: main

text: results,

text: O

text: but

text: we

text: ﬁrst

text: consider

text: the

text: rank

text: subobjects

text: of

text: The

text: rank

text: subobjects

text: S

text: form

text: the

text: base

text: case

text: for

text: our

text: main

text: results,

text: all

text: of

text: which

text: use

text: induction.

text: The

text: case

text: O

text: of

text: is

text: then

text: primarily

text: completed

text: using

text: duality,

text: which

text: allows

text: us

text: to

text: use

text: S

text: O

text: our

text: results

text: for

text: except

text: when

text: D.H

text: S


text: [2] Daniele Arcara, Aaron Bertram, Izzet Coskun, and Jack Huizenga. The P minimal model program for the Hilbert scheme of points on 2 and Bridgeland stability. Adv. Math., 235:580–626, 2013. [3] Arend Bayer and Emanuele Macr`ı. The space of stability conditions on the local projective plane. Duke Math. J., 160:263–322, 2011. [4] Arnaud Beauville. Complex Algebraic Surfaces (London Mathematical Society Student Texts), volume 34. Cambridge University Press, 2nd edition, 1996. [5] Aaron Bertram and Izzet Coskun. The birational geometry of the Hilbert scheme of points on surfaces. In Birational geometry, rational curves and arithmetic, pages 15–55. Springer, 2013. [6] Tom Bridgeland. Stability conditions on triangulated cateogories. Ann. Math., 166:317–345, 2007. [7] Antony Maciocia. Computing the walls associated to Bridgeland stability conditions on projective surfaces. Asian J. Math., 18(2), 2014. [8] Cristian Martinez. Duality, Bridgeland wall-crossing and ﬂips of secant varieties. URL: http://arxiv.org/abs/1311.1183. [9] Yukinobu Toda. Stability conditions and extremal contractions. Math. Ann., 357(2):631–685, 2013. 32

label: reference

text: Daniele

text: Arcara,

text: Aaron

text: Bertram,

text: Izzet

text: Coskun,

text: and

text: Jack

text: Huizenga.

text: The

text: P

text: minimal

text: model

text: program

text: for

text: the

text: Hilbert

text: scheme

text: of

text: points

text: on

text: and

text: Bridgeland

text: stability.

text: Adv.

text: Math.,

text: Arend

text: Bayer

text: and

text: Emanuele

text: Macr`ı.

text: The

text: space

text: of

text: stability

text: conditions

text: on

text: the

text: local

text: projective

text: plane.

text: Duke

text: Math.

text: J.,

text: Arnaud

text: Beauville.

text: Complex

text: Algebraic

text: Surfaces

text: (London

text: Mathematical

text: Society

text: Student

text: Texts),

text: volume

text: Cambridge

text: University

text: Press,

text: edition,

text: Aaron

text: Bertram

text: and

text: Izzet

text: Coskun.

text: The

text: birational

text: geometry

text: of

text: the

text: Hilbert

text: scheme

text: of

text: points

text: on

text: surfaces.

text: In

text: Birational

text: geometry,

text: rational

text: curves

text: and

text: arithmetic,

text: pages

text: Springer,

text: Tom

text: Bridgeland.

text: Stability

text: conditions

text: on

text: triangulated

text: cateogories.

text: Ann.

text: Math.,

text: Antony

text: Maciocia.

text: Computing

text: the

text: walls

text: associated

text: to

text: Bridgeland

text: stability

text: conditions

text: on

text: projective

text: surfaces.

text: Asian

text: J.

text: Math.,

text: Cristian

text: Martinez.

text: Duality,

text: Bridgeland

text: wall-crossing

text: and

text: ﬂips

text: of

text: secant

text: varieties.

text: URL:

text: Yukinobu

text: Toda.

text: Stability

text: conditions

text: and

text: extremal

text: contractions.

text: Math.

text: Ann.,


text: considered only for comparison purposes. lation of the oldest terrains on Vesta, once cor- Even ignoring the roles of Veneneia and rected to obtain the retained population (Marchi Rheasilvia, however, the balance between mass et al., 2012). Our results then imply that the loss and mass gain is markedly in favour of the for- mass loss due to cratering erosion discussed in mer. If, over a certain temporal interval, impacts Sect. g globally aﬀected the surface of the as- were numerous enough to saturate the surface of teroid. The eﬀects of the Late Heavy Bombard- Vesta (as we will discuss in Sect. g), the ero- ment in terms of crater production are equivalent ∼ sive eﬀects associated with cratering could have to another 1 Ga of collisional evolution ( 300 im- removed a signiﬁcant fraction of the previously pacts, S. Pirani, Master Thesis at the University deposited exogenous material. of Rome “La Sapienza”; see also Turrini 2013). 3.7. Crater saturation and blanketing eﬀects Once these craters and the two southern basins As we mentioned previously, from the point of Rheasilvia and Veneneia are added to those pro- view of the observations of the Dawn spacecraft, duced in the following 3.98 Ga, it would appear the blanketing due to the ejecta produced by im- that the crater record on the surface of Vesta can- pacts of non-dark impactors counteracts the de- not allow us to probe much earlier than the Late position of dark material by the dark impactors. Heavy Bombardment itself (as suggested also by Moreover, as we discussed in Sect. g, all impacts Turrini 2013). Given that impacts remove more contribute to the erosion of the surface of Vesta material than they deposit, it is therefore likely and, therefore, to the removal of any previously that the exogenous material delivered to Vesta deposited exogenous material. during its ﬁrst 0.5 Ga of life was mostly stripped In Figure 6 we show the crater populations pro- oﬀ by later impacts. In this case, the bulk of the duced by impacts on Vesta over the last 3.98 Ga dark material we see today on Vesta was brought and the last 1 Ga. In the plot we show also the 5% on the asteroid after the Late Heavy Bombard- and 13% saturation levels: these two threshold ment, justifying our initial choice of focusing on levels represent respectively the minimum value the post-Late Heavy Bombardment timespan. for which a crater population can reach equilib- Even when the crater production rate becomes rium (Melosh, 1989) and the value estimated for low enough not to cause saturation eﬀects, ejecta Mimas, whose surface is the most densely cratered blanketing can still inﬂuence the amount of the in the Solar System (Melosh, 1989). As can be dark material that reside on the surface of Vesta immediately seen, the crater population produced by burying it at diﬀerent depths. We therefore over the last 1 Ga does not cause saturation ef- computed the extension globally covered by ejecta fects. The impacts that Vesta received over the over the two temporal intervals we focused on. last 3.98 Ga, however, produce saturation levels The results are summarised in Table 3, where we comprised between 5% and 13% for crater diam- can see that the cumulative amount of ejecta pro- eters between 50 km and 100 km, and reach the duced over 3.98 Ga would be enough to cover 1.7 13% saturation level between 10 km and 50 km. times the surface of Vesta. The fact that part As discussed by Marchi et al. (2012), the num- of the dark material deposited in the last 3.98 Ga ber of craters retained on the surface of Vesta should be buried by later ejecta provides a natural will be lower than that of the craters produced explanation to the dark veneers observed in crater (see Fig. S2 and the associated discussion in the walls by Jaumann et al. (2012). The ejecta pro- Supplementary Materials of Marchi et al. 2012). duced over the last 1 Ga, on the contrary, would The values we obtained for the crater production aﬀect only 36% of the vestan surface. over the last 3.98 Ga are in broad agreement with We used our model to estimate what could be the theoretical curves computed by Marchi et al. the plausible age of the dark material currently (2012) for old terrains (see Fig. S2 in the Supple- visible on the surface of Vesta. Note that here we mentary Materials of Marchi et al. 2012), which are referring only to the dark material that is ob- in turn are in agreement with the crater popu- served by the Framing Camera and the VIR spec- 19

text: considered

text: only

text: for

text: comparison

text: purposes.

text: lation

text: of

text: the

text: oldest

text: terrains

text: on

text: Vesta,

text: once

text: cor-

text: Even

text: ignoring

text: the

text: roles

text: of

text: Veneneia

text: and

text: rected

text: to

text: obtain

text: the

text: retained

text: population

text: (Marchi

text: Rheasilvia,

text: however,

text: the

text: balance

text: between

text: mass

text: et

text: al.,

text: Our

text: results

text: then

text: imply

text: that

text: the

text: loss

text: and

text: mass

text: gain

text: is

text: markedly

text: in

text: favour

text: of

text: the

text: for-

text: mass

text: loss

text: due

text: to

text: cratering

text: erosion

text: discussed

text: in

text: mer.

text: If,

text: over

text: a

text: certain

text: temporal

text: interval,

text: impacts

text: Sect.

text: g

text: globally

text: aﬀected

text: the

text: surface

text: of

text: the

text: as-

text: were

text: numerous

text: enough

text: to

text: saturate

text: the

text: surface

text: of

text: teroid.

text: The

text: eﬀects

text: of

text: the

text: Late

text: Heavy

text: Bombard-

text: Vesta

text: (as

text: we

text: will

text: discuss

text: in

text: Sect.

text: g),

text: the

text: ero-

text: ment

text: in

text: terms

text: of

text: crater

text: production

text: are

text: equivalent

text: sive

text: eﬀects

text: associated

text: with

text: cratering

text: could

text: have

text: to

text: another

text: Ga

text: of

text: collisional

text: evolution

text: im-

text: removed

text: a

text: signiﬁcant

text: fraction

text: of

text: the

text: previously

text: pacts,

text: S.

text: Pirani,

text: Master

text: Thesis

text: at

text: the

text: University

text: deposited

text: exogenous

text: material.

text: of

text: Rome

text: “La

text: Sapienza”;

text: see

text: also

text: Turrini

text: Crater

text: saturation

text: and

text: blanketing

text: eﬀects

text: Once

text: these

text: craters

text: and

text: the

text: two

text: southern

text: basins

text: As

text: we

text: mentioned

text: previously,

text: from

text: the

text: point

text: of

text: Rheasilvia

text: and

text: Veneneia

text: are

text: added

text: to

text: those

text: pro-

text: view

text: of

text: the

text: observations

text: of

text: the

text: Dawn

text: spacecraft,

text: duced

text: in

text: the

text: following

text: Ga,

text: it

text: would

text: appear

text: the

text: blanketing

text: due

text: to

text: the

text: ejecta

text: produced

text: by

text: im-

text: that

text: the

text: crater

text: record

text: on

text: the

text: surface

text: of

text: Vesta

text: can-

text: pacts

text: of

text: non-dark

text: impactors

text: counteracts

text: the

text: de-

text: not

text: allow

text: us

text: to

text: probe

text: much

text: earlier

text: than

text: the

text: Late

text: position

text: of

text: dark

text: material

text: by

text: the

text: dark

text: impactors.

text: Heavy

text: Bombardment

text: itself

text: (as

text: suggested

text: also

text: by

text: Moreover,

text: as

text: we

text: discussed

text: in

text: Sect.

text: g,

text: all

text: impacts

text: Turrini

text: Given

text: that

text: impacts

text: remove

text: more

text: contribute

text: to

text: the

text: erosion

text: of

text: the

text: surface

text: of

text: Vesta

text: material

text: than

text: they

text: deposit,

text: it

text: is

text: therefore

text: likely

text: and,

text: therefore,

text: to

text: the

text: removal

text: of

text: any

text: previously

text: that

text: the

text: exogenous

text: material

text: delivered

text: to

text: Vesta

text: deposited

text: exogenous

text: material.

text: during

text: its

text: ﬁrst

text: Ga

text: of

text: life

text: was

text: mostly

text: stripped

text: In

text: Figure

text: we

text: show

text: the

text: crater

text: populations

text: pro-

text: oﬀ

text: by

text: later

text: impacts.

text: In

text: this

text: case,

text: the

text: bulk

text: of

text: the

text: duced

text: by

text: impacts

text: on

text: Vesta

text: over

text: the

text: last

text: Ga

text: dark

text: material

text: we

text: see

text: today

text: on

text: Vesta

text: was

text: brought

text: and

text: the

text: last

text: Ga.

text: In

text: the

text: plot

text: we

text: show

text: also

text: the

text: on

text: the

text: asteroid

text: after

text: the

text: Late

text: Heavy

text: Bombard-

text: and

text: saturation

text: levels:

text: these

text: two

text: threshold

text: ment,

text: justifying

text: our

text: initial

text: choice

text: of

text: focusing

text: on

text: levels

text: represent

text: respectively

text: the

text: minimum

text: value

text: the

text: post-Late

text: Heavy

text: Bombardment

text: timespan.

text: for

text: which

text: a

text: crater

text: population

text: can

text: reach

text: equilib-

text: Even

text: when

text: the

text: crater

text: production

text: rate

text: becomes

text: rium

text: (Melosh,

text: and

text: the

text: value

text: estimated

text: for

text: low

text: enough

text: not

text: to

text: cause

text: saturation

text: eﬀects,

text: ejecta

text: Mimas,

text: whose

text: surface

text: is

text: the

text: most

text: densely

text: cratered

text: blanketing

text: can

text: still

text: inﬂuence

text: the

text: amount

text: of

text: the

text: in

text: the

text: Solar

text: System

text: (Melosh,

text: As

text: can

text: be

text: dark

text: material

text: that

text: reside

text: on

text: the

text: surface

text: of

text: Vesta

text: immediately

text: seen,

text: the

text: crater

text: population

text: produced

text: by

text: burying

text: it

text: at

text: diﬀerent

text: depths.

text: We

text: therefore

text: over

text: the

text: last

text: Ga

text: does

text: not

text: cause

text: saturation

text: ef-

text: computed

text: the

text: extension

text: globally

text: covered

text: by

text: ejecta

text: fects.

text: The

text: impacts

text: that

text: Vesta

text: received

text: over

text: the

text: over

text: the

text: two

text: temporal

text: intervals

text: we

text: focused

text: on.

text: last

text: Ga,

text: however,

text: produce

text: saturation

text: levels

text: The

text: results

text: are

text: summarised

text: in

text: Table

text: where

text: we

text: comprised

text: between

text: and

text: for

text: crater

text: diam-

text: can

text: see

text: that

text: the

text: cumulative

text: amount

text: of

text: ejecta

text: pro-

text: eters

text: between

text: km

text: and

text: km,

text: and

text: reach

text: the

text: duced

text: over

text: Ga

text: would

text: be

text: enough

text: to

text: cover

text: saturation

text: level

text: between

text: km

text: and

text: km.

text: times

text: the

text: surface

text: of

text: Vesta.

text: The

text: fact

text: that

text: part

text: As

text: discussed

text: by

text: Marchi

text: et

text: al.

text: the

text: num-

text: of

text: the

text: dark

text: material

text: deposited

text: in

text: the

text: last

text: Ga

text: ber

text: of

text: craters

text: retained

text: on

text: the

text: surface

text: of

text: Vesta

text: should

text: be

text: buried

text: by

text: later

text: ejecta

text: provides

text: a

text: natural

text: will

text: be

text: lower

text: than

text: that

text: of

text: the

text: craters

text: produced

text: explanation

text: to

text: the

text: dark

text: veneers

text: observed

text: in

text: crater

text: (see

text: Fig.

text: and

text: the

text: associated

text: discussion

text: in

text: the

text: walls

text: by

text: Jaumann

text: et

text: al.

text: The

text: ejecta

text: pro-

text: Supplementary

text: Materials

text: of

text: Marchi

text: et

text: al.

text: duced

text: over

text: the

text: last

text: Ga,

text: on

text: the

text: contrary,

text: would

text: The

text: values

text: we

text: obtained

text: for

text: the

text: crater

text: production

text: aﬀect

text: only

text: of

text: the

text: vestan

text: surface.

text: over

text: the

text: last

text: Ga

text: are

text: in

text: broad

text: agreement

text: with

text: We

text: used

text: our

text: model

text: to

text: estimate

text: what

text: could

text: be

text: the

text: theoretical

text: curves

text: computed

text: by

text: Marchi

text: et

text: al.

text: the

text: plausible

text: age

text: of

text: the

text: dark

text: material

text: currently

text: for

text: old

text: terrains

text: (see

text: Fig.

text: in

text: the

text: Supple-

text: visible

text: on

text: the

text: surface

text: of

text: Vesta.

text: Note

text: that

text: here

text: we

text: mentary

text: Materials

text: of

text: Marchi

text: et

text: al.

text: which

text: are

text: referring

text: only

text: to

text: the

text: dark

text: material

text: that

text: is

text: ob-

text: in

text: turn

text: are

text: in

text: agreement

text: with

text: the

text: crater

text: popu-

text: served

text: by

text: the

text: Framing

text: Camera

text: and

text: the

text: VIR

text: spec-


text: trometer on-board the Dawn spacecraft, which ∼ are sensitive to the topmost 1 cm of the vestan regolith. The contaminants mixed in the topmost 1 m of the vestan regolith are instead measured by GRaND and will be discussed in Sect. g and Sect. g. As we showed in Table 3, impacts over the last 2.6 Ga produce enough ejecta to cover a surface equal to that of Vesta. The dark ma- terial deposited from 2.6 Ga ago to now should, on average, remain exposed on the surface of the asteroid, conﬁrming the prediction we made in the Supplementary Information of McCord et al. (2012). The surface coverage due to the dark ma- terial younger than 2.6 Ga should be of the order of 14% of the vestan surface. The values of the surface coverage here esti- mated should be corrected, however, for the still undetermined eﬀects of the fraction of buried dark material that was exposed by later impacts and of the dark material mixed in the regolith by the micrometeoritic ﬂux. Nevertheless, in terms of or- ders of magnitude the darkened surface visible to ∼ date on Vesta should vary between 10%, which is the same order of magnitude of the darkest re- gions in Figure 4 (R < 0.26 or, equivalently, up ∼ to the magenta region), and 30%, which is the Figure 6: Size-frequency distribution of the crater popu- same value estimated by GRaND for the exten- lation produced on Vesta by all impactors over the last 3.98 Ga and the last 1 Ga. The 5% and 13% saturation sion of the most H-rich (H concentration equal to levels are shown for reference. The eﬀects of Veneneia and or higher than 250 µg/g, Prettyman et al. 2012) Rheasilvia are not included in this plot. regions. Note that, if we compute the crater population produced over the last 2.6 Ga and plot the results picture of the survival of the dark material ex- in the same way as Figure 6, the population of posed on the vestan surface or buried in the ves- ∼ craters below 70 km will surpass the 5% satu- tan regolith, we can now proceed to quantitatively ration level. It is therefore possible that craters estimate the amounts of dark material and water- produced over the last 2.6 Ga removed part of the equivalent material delivered since the Late Heavy dark material already on the surface. This could Bombardment and compare them with the mea- explain the existence of the dark spots not asso- surements supplied by the GRaND instrument on- ciated with craters observed on Vesta (Reddy et board the Dawn spacecraft. al., 2012). The relevant craters could have been If we focus only on the size bins from Table 1 removed by later impacts leaving only the dark that are associated with at least 1 impact event material deposited outside their borders, e.g. in respectively of dark, non-dark and all kind of im- the downstream region respect to the impact di- pactors over the desired timespan (see e.g. Figure rection (Artemieva and Pierazzo, 2011). 2 for the case of the last 3.98 Ga), we can use the 3.8. Contamination by dark and water-equivalent ﬂuxes of impactors we computed to estimate the materials amount of the diﬀerent exogenous materials deliv- Since the previous results provided us a clearer ered to Vesta (see e.g. Figure 3 for the case of the 20 ##LTFigure##

text: trometer

text: on-board

text: the

text: Dawn

text: spacecraft,

text: which

text: are

text: sensitive

text: to

text: the

text: topmost

text: cm

text: of

text: the

text: vestan

text: regolith.

text: The

text: contaminants

text: mixed

text: in

text: the

text: topmost

text: m

text: of

text: the

text: vestan

text: regolith

text: are

text: instead

text: measured

text: by

text: GRaND

text: and

text: will

text: be

text: discussed

text: in

text: Sect.

text: g

text: and

text: Sect.

text: g.

text: As

text: we

text: showed

text: in

text: Table

text: impacts

text: over

text: the

text: last

text: Ga

text: produce

text: enough

text: ejecta

text: to

text: cover

text: a

text: surface

text: equal

text: to

text: that

text: of

text: Vesta.

text: The

text: dark

text: ma-

text: terial

text: deposited

text: from

text: Ga

text: ago

text: to

text: now

text: should,

text: on

text: average,

text: remain

text: exposed

text: on

text: the

text: surface

text: of

text: the

text: asteroid,

text: conﬁrming

text: the

text: prediction

text: we

text: made

text: in

text: the

text: Supplementary

text: Information

text: of

text: McCord

text: et

text: al.

text: The

text: surface

text: coverage

text: due

text: to

text: the

text: dark

text: ma-

text: terial

text: younger

text: than

text: Ga

text: should

text: be

text: of

text: the

text: order

text: of

text: of

text: the

text: vestan

text: surface.

text: The

text: values

text: of

text: the

text: surface

text: coverage

text: here

text: esti-

text: mated

text: should

text: be

text: corrected,

text: however,

text: for

text: the

text: still

text: undetermined

text: eﬀects

text: of

text: the

text: fraction

text: of

text: buried

text: dark

text: material

text: that

text: was

text: exposed

text: by

text: later

text: impacts

text: and

text: of

text: the

text: dark

text: material

text: mixed

text: in

text: the

text: regolith

text: by

text: the

text: micrometeoritic

text: ﬂux.

text: Nevertheless,

text: in

text: terms

text: of

text: or-

text: ders

text: of

text: magnitude

text: the

text: darkened

text: surface

text: visible

text: to

text: date

text: on

text: Vesta

text: should

text: vary

text: between

text: which

text: is

text: the

text: same

text: order

text: of

text: magnitude

text: of

text: the

text: darkest

text: re-

text: gions

text: in

text: Figure

text: (R

text: or,

text: equivalently,

text: up

text: to

text: the

text: magenta

text: region),

text: and

text: which

text: is

text: the

text: Figure

text: Size-frequency

text: distribution

text: of

text: the

text: crater

text: popu-

text: same

text: value

text: estimated

text: by

text: GRaND

text: for

text: the

text: exten-

text: lation

text: produced

text: on

text: Vesta

text: by

text: all

text: impactors

text: over

text: the

text: last

text: Ga

text: and

text: the

text: last

text: Ga.

text: The

text: and

text: saturation

text: sion

text: of

text: the

text: most

text: H-rich

text: (H

text: concentration

text: equal

text: to

text: levels

text: are

text: shown

text: for

text: reference.

text: The

text: eﬀects

text: of

text: Veneneia

text: and

text: or

text: higher

text: than

text: µg/g,

text: Prettyman

text: et

text: al.

text: Rheasilvia

text: are

text: not

text: included

text: in

text: this

text: plot.

text: regions.

text: Note

text: that,

text: if

text: we

text: compute

text: the

text: crater

text: population

text: produced

text: over

text: the

text: last

text: Ga

text: and

text: plot

text: the

text: results

text: picture

text: of

text: the

text: survival

text: of

text: the

text: dark

text: material

text: ex-

text: in

text: the

text: same

text: way

text: as

text: Figure

text: the

text: population

text: of

text: posed

text: on

text: the

text: vestan

text: surface

text: or

text: buried

text: in

text: the

text: ves-

text: craters

text: below

text: km

text: will

text: surpass

text: the

text: satu-

text: tan

text: regolith,

text: we

text: can

text: now

text: proceed

text: to

text: quantitatively

text: ration

text: level.

text: It

text: is

text: therefore

text: possible

text: that

text: craters

text: estimate

text: the

text: amounts

text: of

text: dark

text: material

text: and

text: water-

text: produced

text: over

text: the

text: last

text: Ga

text: removed

text: part

text: of

text: the

text: equivalent

text: material

text: delivered

text: since

text: the

text: Late

text: Heavy

text: dark

text: material

text: already

text: on

text: the

text: surface.

text: This

text: could

text: Bombardment

text: and

text: compare

text: them

text: with

text: the

text: mea-

text: explain

text: the

text: existence

text: of

text: the

text: dark

text: spots

text: not

text: asso-

text: surements

text: supplied

text: by

text: the

text: GRaND

text: instrument

text: on-

text: ciated

text: with

text: craters

text: observed

text: on

text: Vesta

text: (Reddy

text: et

text: board

text: the

text: Dawn

text: spacecraft.

text: al.,

text: The

text: relevant

text: craters

text: could

text: have

text: been

text: If

text: we

text: focus

text: only

text: on

text: the

text: size

text: bins

text: from

text: Table

text: removed

text: by

text: later

text: impacts

text: leaving

text: only

text: the

text: dark

text: that

text: are

text: associated

text: with

text: at

text: least

text: impact

text: event

text: material

text: deposited

text: outside

text: their

text: borders,

text: e.g.

text: in

text: respectively

text: of

text: dark,

text: non-dark

text: and

text: all

text: kind

text: of

text: im-

text: the

text: downstream

text: region

text: respect

text: to

text: the

text: impact

text: di-

text: pactors

text: over

text: the

text: desired

text: timespan

text: (see

text: e.g.

text: Figure

text: rection

text: (Artemieva

text: and

text: Pierazzo,

text: for

text: the

text: case

text: of

text: the

text: last

text: Ga),

text: we

text: can

text: use

text: the

text: Contamination

text: by

text: dark

text: and

text: water-equivalent

text: ﬂuxes

text: of

text: impactors

text: we

text: computed

text: to

text: estimate

text: the

text: materials

text: amount

text: of

text: the

text: diﬀerent

text: exogenous

text: materials

text: deliv-

text: Since

text: the

text: previous

text: results

text: provided

text: us

text: a

text: clearer

text: ered

text: to

text: Vesta

text: (see

text: e.g.

text: Figure

text: for

text: the

text: case

text: of

text: the

text: ##LTFigure##


text: signiﬁcant role in the contamination of the surface low albedo impactors on Vesta computing the ra- of the asteroid (see e.g. Turrini et al. 2011), we fo- tio between asteroids with B, C, P and D spectral cused only on the single value of the average reten- types and all spectroscopically classiﬁed asteroids tion eﬃciency reported for asteroidal impactors. from the JPL Small-Body Database Search En- To estimate the diﬀerence between the results of gine2. The value they obtained was F = 0.22. d Ong et al. (2010) and those of Svetsov (2011), An independent estimate performed by De Sanc- ≤ ≤ we computed the average retained mass fraction tis et al. (2012a) consistently gave 0.1 F 0.3. d for the Moon using the scaling laws reported by In this work, we used the catalogue of aster- Svetsov (2011). Following Ong et al. (2010), we oids from the JPL Small-Body Database Search used an escape velocity of 2.38 km/s and an aver- Engine to assess the fraction of low albedo im- age impact velocity of 20 km/s. The logarithmic pactors on Vesta in three diﬀerent ways, i.e.: 1) interpolation of Eqs. 8 and 9 from Svetsov (2011) using the same method as McCord et al. (2012); for the chosen impact velocity gives an average re- 2) using the same method as McCord et al. (2012) tained mass of 55.7%. Therefore, the scaling laws but including only asteroids with C, D and P spec- by Svetsov (2011) result in a retention eﬃciency tral types; 3) computing the fraction of asteroids about 3.4 times larger than the results of Ong et with albedo lower than or equal to 0.05 among al. (2010). all asteroids for which a value of albedo has been Before proceeding, it must be stressed that lin- estimated. As mentioned above, method 1 held early scaling Eq. 3 to the results of Ong et al. a value of F = 0.22. Method 2 held a value of d1 (2010) obtained for an average impact velocity of F = 0.18. Method 3 held a value of F = 0.28. d2 d3 20 km/s is not necessarily correct over the range The number of dark impactors of a given size D − i of impact velocities characteristic on Vesta (1 10 over a temporal interval ∆T on Vesta is then km/s, O’Brien and Sykes 2011). However, as we × are mainly interested in understanding how much ndark = n F (5) i i d the uncertainty on the retention scaling law can aﬀect the results of the model, in the following we where n is obtained though Eq. 2 and F is i d will also consider the following case: one of the three values previously discussed. The amount of dark m(cid:16)aterial(cid:17)delivered by these im- ∗ pactors can instead be expressed as f = 0.296 f r ×r = 0.296 ((0.14 + 0.003V ) (4) π × × × ln v + 0.9V −0.24) mdiark = 6 Di3ρ fr ndiark (6) esc where f is obtained either through Eq. 3 or Using the same escape velocity and average im- r Eq. 4 and ρ is the average density of the dark pact speed we used for Eq. 3, Eq. 4 gives a re- ∗ impactors. In the following we will use the value tention eﬃciency f = 0.138. The retention eﬃ- r derived by McCord et al. (2012) and through our ciencies supplied by Eqs. 3 and 4 will be used to method 1 as our reference value. We will use the transform the ﬂuxes of impactors (computed as results of methods 2 and 3 to discuss how the described in Sect. g) into mass ﬂuxes on Vesta. uncertainty on the abundance of dark asteroids in Using the abundances of the diﬀerent kind of im- the asteroids belt aﬀects the results of the model. pactors and their compositions, which we will de- Coincident with that of the dark material, the scribe in Sects. g and g, we will then use to the Dawn mission reported the presence of OH (De mass ﬂuxes thus obtained to estimate the contam- Sanctis et al., 2012a) and H-rich material (Pret- ination of the surface of Vesta by diﬀerent classes tyman et al., 2012) on Vesta. The results of Pret- of exogenous materials. tyman et al. (2012) and De Sanctis et al. (2012a) 2.4. Dark impactors, OH and H-rich material McCord et al. (2012) estimated the fraction of 2http://ssd.jpl.nasa.gov/sbdb_query.cgi 7 ##LTLine## ##LTLine##

text: signiﬁcant

text: role

text: in

text: the

text: contamination

text: of

text: the

text: surface

text: low

text: albedo

text: impactors

text: on

text: Vesta

text: computing

text: the

text: ra-

text: of

text: the

text: asteroid

text: (see

text: e.g.

text: Turrini

text: et

text: al.

text: we

text: fo-

text: tio

text: between

text: asteroids

text: with

text: B,

text: C,

text: P

text: and

text: D

text: spectral

text: cused

text: only

text: on

text: the

text: single

text: value

text: of

text: the

text: average

text: reten-

text: types

text: and

text: all

text: spectroscopically

text: classiﬁed

text: asteroids

text: tion

text: eﬃciency

text: reported

text: for

text: asteroidal

text: impactors.

text: from

text: the

text: JPL

text: Small-Body

text: Database

text: Search

text: En-

text: To

text: estimate

text: the

text: diﬀerence

text: between

text: the

text: results

text: of

text: The

text: value

text: they

text: obtained

text: was

text: F

text: d

text: Ong

text: et

text: al.

text: and

text: those

text: of

text: Svetsov

text: An

text: independent

text: estimate

text: performed

text: by

text: De

text: Sanc-

text: we

text: computed

text: the

text: average

text: retained

text: mass

text: fraction

text: tis

text: et

text: al.

text: consistently

text: gave

text: F

text: d

text: for

text: the

text: Moon

text: using

text: the

text: scaling

text: laws

text: reported

text: by

text: In

text: this

text: work,

text: we

text: used

text: the

text: catalogue

text: of

text: aster-

text: Svetsov

text: Following

text: Ong

text: et

text: al.

text: we

text: oids

text: from

text: the

text: JPL

text: Small-Body

text: Database

text: Search

text: used

text: an

text: escape

text: velocity

text: of

text: km/s

text: and

text: an

text: aver-

text: Engine

text: to

text: assess

text: the

text: fraction

text: of

text: low

text: albedo

text: im-

text: age

text: impact

text: velocity

text: of

text: km/s.

text: The

text: logarithmic

text: pactors

text: on

text: Vesta

text: in

text: three

text: diﬀerent

text: ways,

text: i.e.:

text: interpolation

text: of

text: Eqs.

text: and

text: from

text: Svetsov

text: using

text: the

text: same

text: method

text: as

text: McCord

text: et

text: al.

text: for

text: the

text: chosen

text: impact

text: velocity

text: gives

text: an

text: average

text: re-

text: using

text: the

text: same

text: method

text: as

text: McCord

text: et

text: al.

text: tained

text: mass

text: of

text: Therefore,

text: the

text: scaling

text: laws

text: but

text: including

text: only

text: asteroids

text: with

text: C,

text: D

text: and

text: P

text: spec-

text: by

text: Svetsov

text: result

text: in

text: a

text: retention

text: eﬃciency

text: tral

text: types;

text: computing

text: the

text: fraction

text: of

text: asteroids

text: about

text: times

text: larger

text: than

text: the

text: results

text: of

text: Ong

text: et

text: with

text: albedo

text: lower

text: than

text: or

text: equal

text: to

text: among

text: al.

text: all

text: asteroids

text: for

text: which

text: a

text: value

text: of

text: albedo

text: has

text: been

text: Before

text: proceeding,

text: it

text: must

text: be

text: stressed

text: that

text: lin-

text: estimated.

text: As

text: mentioned

text: above,

text: method

text: held

text: early

text: scaling

text: Eq.

text: to

text: the

text: results

text: of

text: Ong

text: et

text: al.

text: a

text: value

text: of

text: F

text: Method

text: held

text: a

text: value

text: of

text: obtained

text: for

text: an

text: average

text: impact

text: velocity

text: of

text: F

text: Method

text: held

text: a

text: value

text: of

text: F

text: km/s

text: is

text: not

text: necessarily

text: correct

text: over

text: the

text: range

text: The

text: number

text: of

text: dark

text: impactors

text: of

text: a

text: given

text: size

text: D

text: i

text: of

text: impact

text: velocities

text: characteristic

text: on

text: Vesta

text: over

text: a

text: temporal

text: interval

text: ∆T

text: on

text: Vesta

text: is

text: then

text: km/s,

text: O’Brien

text: and

text: Sykes

text: However,

text: as

text: we

text: are

text: mainly

text: interested

text: in

text: understanding

text: how

text: much

text: ndark

text: n

text: F

text: i

text: i

text: d

text: the

text: uncertainty

text: on

text: the

text: retention

text: scaling

text: law

text: can

text: aﬀect

text: the

text: results

text: of

text: the

text: model,

text: in

text: the

text: following

text: we

text: where

text: n

text: is

text: obtained

text: though

text: Eq.

text: and

text: F

text: is

text: i

text: d

text: will

text: also

text: consider

text: the

text: following

text: case:

text: one

text: of

text: the

text: three

text: values

text: previously

text: discussed.

text: The

text: amount

text: of

text: dark

text: by

text: these

text: im-

text: pactors

text: can

text: instead

text: be

text: expressed

text: as

text: f

text: f

text: r

text: ×r

text: π

text: ln

text: v

text: mdiark

text: fr

text: ndiark

text: esc

text: where

text: f

text: is

text: obtained

text: either

text: through

text: Eq.

text: or

text: Using

text: the

text: same

text: escape

text: velocity

text: and

text: average

text: im-

text: r

text: Eq.

text: and

text: ρ

text: is

text: the

text: average

text: density

text: of

text: the

text: dark

text: pact

text: speed

text: we

text: used

text: for

text: Eq.

text: Eq.

text: gives

text: a

text: re-

text: impactors.

text: In

text: the

text: following

text: we

text: will

text: use

text: the

text: value

text: tention

text: eﬃciency

text: f

text: The

text: retention

text: eﬃ-

text: r

text: derived

text: by

text: McCord

text: et

text: al.

text: and

text: through

text: our

text: ciencies

text: supplied

text: by

text: Eqs.

text: and

text: will

text: be

text: used

text: to

text: method

text: as

text: our

text: reference

text: value.

text: We

text: will

text: use

text: the

text: transform

text: the

text: ﬂuxes

text: of

text: impactors

text: (computed

text: as

text: results

text: of

text: methods

text: and

text: to

text: discuss

text: how

text: the

text: described

text: in

text: Sect.

text: g)

text: into

text: mass

text: ﬂuxes

text: on

text: Vesta.

text: uncertainty

text: on

text: the

text: abundance

text: of

text: dark

text: asteroids

text: in

text: Using

text: the

text: abundances

text: of

text: the

text: diﬀerent

text: kind

text: of

text: im-

text: the

text: asteroids

text: belt

text: aﬀects

text: the

text: results

text: of

text: the

text: model.

text: pactors

text: and

text: their

text: compositions,

text: which

text: we

text: will

text: de-

text: Coincident

text: with

text: that

text: of

text: the

text: dark

text: material,

text: the

text: scribe

text: in

text: Sects.

text: g

text: and

text: g,

text: we

text: will

text: then

text: use

text: to

text: the

text: Dawn

text: mission

text: reported

text: the

text: presence

text: of

text: OH

text: (De

text: mass

text: ﬂuxes

text: thus

text: obtained

text: to

text: estimate

text: the

text: contam-

text: Sanctis

text: et

text: al.,

text: and

text: H-rich

text: material

text: (Pret-

text: ination

text: of

text: the

text: surface

text: of

text: Vesta

text: by

text: diﬀerent

text: classes

text: tyman

text: et

text: al.,

text: on

text: Vesta.

text: The

text: results

text: of

text: Pret-

text: of

text: exogenous

text: materials.

text: tyman

text: et

text: al.

text: and

text: De

text: Sanctis

text: et

text: al.

text: Dark

text: impactors,

text: OH

text: and

text: H-rich

text: material

text: McCord

text: et

text: al.

text: estimated

text: the

text: fraction

text: of

text: ##LTLine##

text: ##LTLine##


text: indicated a clear correlation between the presence If, following Zolensky et al. (1996), we focus of dark material and that of OH and H-rich ma- only on CM and CR chondrites, we obtain that terial. Both H and OH are likely present in the a fraction F CM+CR = 0.41 of the dark impactors w vestan regolith in the form of hydrated minerals on Vesta would bring water-equivalent material and not as water. However, in meteoritics (see to the asteroid. This will be our standard case. e.g. Robert 2003) and in nuclear spectroscopy Should we include also CI chondrites, i.e. we as- (see e.g. Prettyman 2007) their presence is gener- sume that our database of carbonaceous clasts in ally quantiﬁed in terms of the equivalent amount HED meteorites is not complete, the fraction of of water needed to reproduce the measurements. carriers of water-equivalent material would rise In the following, therefore, we will refer to these to F CM+CR+CI = 0.53. We caution the readers, w two (likely overlapping) classes of materials with however, that while this is possibly the best ap- the common term of “water-equivalent material.” proximation we can make to date, it is not nec- As discussed by McCord et al. (2012), the dark essarily a good or physically meaningful one. As impactors are plausibly asteroids similar in com- an example, CI chondrites are very fragile and position to the carbonaceous chondrites. The the bulk of them likely do not survive the pas- most eﬃcient carriers of water-equivalent mate- sage through the Earth’s atmosphere, causing the rial among the dark impactors should therefore underestimation of their contribution. Moreover, be those bodies similar in composition to the CM, there is no guarantee that the modern ﬂux of me- CR and CI carbonaceous chondrites (Jarosewich, teorites on Earth is a meaningful approximation 1990; Robert, 2003). In particular, CM chondrites of the secular ﬂux in the asteroid belt over one and, to a smaller extent, CR chondrites represent or more Ga, as it can be signiﬁcantly aﬀected by the dominant components of carbonaceous chon- temporally-limited events like the break-up of an dritic clasts observed in HED meteorites (Zolen- asteroid. sky et al., 1996; Lorenz et al., 2007). The CR/CM In order to better constrain the implications of ratio in HED meteorites, moreover, is the same as the uncertainty on the ﬂux of carriers of water- the one observed among modern CM and CR falls equivalent material for our results, we consid- on Earth (Zolensky et al., 1996). Therefore, in or- ered also the water delivery eﬃciency discussed der to assess the amount of water-equivalent ma- by Ong et al. (2010). In their work, these au- terial that a continuous ﬂux of impactors would thors assumed that about one-third of impacting bring on Vesta, we tried to estimate the frequency asteroids would be C-type (our dark impactors, of CM-, CR- and CI-like projectiles among the i.e. F = 0.33) and that two-thirds of those im- d dark impactors hitting the asteroid. Based on pactors would contain water-equivalent material the observation by Zolensky et al. (1996) that (F = 0.67). These authors assumed an average w the CR/CM ratios are similar among meteorite 10 wt% water content, i.e. the same value we falls on Earth and HEDs meteorites, we used the adopted in the previous cases. As these authors ﬂuxes of the falls for the diﬀerent classes of car- point out, their assumptions result in water repre- bonaceous chondrites on Earth as our planetary senting 2.2% of the total mass ﬂux due to impacts. analogue. From the Meteoritical Bullettin3, as As a comparison, in our reference case we have × × of May 2013, we note that CM chondrites repre- F F CM+CR R = 0.9%, i.e. our reference d1 w w sent 34% of the falls of carbonaceous chondrites, water delivery eﬃciency is 2.5 times smaller. CR chondrites represent 6.8% and CI chondrites Using the value of R and one of the possible w 11.4%. The fraction of the masses of these im- values of F previously discussed, the amount of w pactors that represent water-equivalent material water-equivalent material mw delivered on Vesta i can be assumed, on average, R = 0.1, i.e. 10 by dark impactors of a given size D over a tem- w i wt% (Jarosewich, 1990; Robert, 2003). poral interval ∆T is then × × 3http://www.lpi.usra.edu/meteor mwi = mdiark Fw Rw (7) 8 ##LTLine##

text: indicated

text: a

text: clear

text: correlation

text: between

text: the

text: presence

text: If,

text: following

text: Zolensky

text: et

text: al.

text: we

text: focus

text: of

text: dark

text: material

text: and

text: that

text: of

text: OH

text: and

text: H-rich

text: ma-

text: only

text: on

text: CM

text: and

text: CR

text: chondrites,

text: we

text: obtain

text: that

text: terial.

text: Both

text: H

text: and

text: OH

text: are

text: likely

text: present

text: in

text: the

text: a

text: fraction

text: F

text: CM+CR

text: of

text: the

text: dark

text: impactors

text: w

text: vestan

text: regolith

text: in

text: the

text: form

text: of

text: hydrated

text: minerals

text: on

text: Vesta

text: would

text: bring

text: water-equivalent

text: material

text: and

text: not

text: as

text: water.

text: However,

text: in

text: meteoritics

text: (see

text: to

text: the

text: asteroid.

text: This

text: will

text: be

text: our

text: standard

text: case.

text: e.g.

text: Robert

text: and

text: in

text: nuclear

text: spectroscopy

text: Should

text: we

text: include

text: also

text: CI

text: chondrites,

text: i.e.

text: we

text: as-

text: (see

text: e.g.

text: Prettyman

text: their

text: presence

text: is

text: gener-

text: sume

text: that

text: our

text: database

text: of

text: carbonaceous

text: clasts

text: in

text: ally

text: quantiﬁed

text: in

text: terms

text: of

text: the

text: equivalent

text: amount

text: HED

text: meteorites

text: is

text: not

text: complete,

text: the

text: fraction

text: of

text: of

text: water

text: needed

text: to

text: reproduce

text: the

text: measurements.

text: carriers

text: of

text: water-equivalent

text: material

text: would

text: rise

text: In

text: the

text: following,

text: therefore,

text: we

text: will

text: refer

text: to

text: these

text: to

text: F

text: CM+CR+CI

text: We

text: caution

text: the

text: readers,

text: w

text: two

text: (likely

text: overlapping)

text: classes

text: of

text: materials

text: with

text: however,

text: that

text: while

text: this

text: is

text: possibly

text: the

text: best

text: ap-

text: the

text: common

text: term

text: of

text: “water-equivalent

text: material.”

text: proximation

text: we

text: can

text: make

text: to

text: date,

text: it

text: is

text: not

text: nec-

text: As

text: discussed

text: by

text: McCord

text: et

text: al.

text: the

text: dark

text: essarily

text: a

text: good

text: or

text: physically

text: meaningful

text: one.

text: As

text: impactors

text: are

text: plausibly

text: asteroids

text: similar

text: in

text: com-

text: an

text: example,

text: CI

text: chondrites

text: are

text: very

text: fragile

text: and

text: position

text: to

text: the

text: carbonaceous

text: chondrites.

text: The

text: the

text: bulk

text: of

text: them

text: likely

text: do

text: not

text: survive

text: the

text: pas-

text: most

text: eﬃcient

text: carriers

text: of

text: water-equivalent

text: mate-

text: sage

text: through

text: the

text: Earth’s

text: atmosphere,

text: causing

text: the

text: rial

text: among

text: the

text: dark

text: impactors

text: should

text: therefore

text: underestimation

text: of

text: their

text: contribution.

text: Moreover,

text: be

text: those

text: bodies

text: similar

text: in

text: composition

text: to

text: the

text: CM,

text: there

text: is

text: no

text: guarantee

text: that

text: the

text: modern

text: ﬂux

text: of

text: me-

text: CR

text: and

text: CI

text: carbonaceous

text: chondrites

text: (Jarosewich,

text: teorites

text: on

text: Earth

text: is

text: a

text: meaningful

text: approximation

text: Robert,

text: In

text: particular,

text: CM

text: chondrites

text: of

text: the

text: secular

text: ﬂux

text: in

text: the

text: asteroid

text: belt

text: over

text: one

text: and,

text: to

text: a

text: smaller

text: extent,

text: CR

text: chondrites

text: represent

text: or

text: more

text: Ga,

text: as

text: it

text: can

text: be

text: signiﬁcantly

text: aﬀected

text: by

text: the

text: dominant

text: components

text: of

text: carbonaceous

text: chon-

text: temporally-limited

text: events

text: like

text: the

text: break-up

text: of

text: an

text: dritic

text: clasts

text: observed

text: in

text: HED

text: meteorites

text: (Zolen-

text: asteroid.

text: sky

text: et

text: al.,

text: Lorenz

text: et

text: al.,

text: The

text: CR/CM

text: In

text: order

text: to

text: better

text: constrain

text: the

text: implications

text: of

text: ratio

text: in

text: HED

text: meteorites,

text: moreover,

text: is

text: the

text: same

text: as

text: the

text: uncertainty

text: on

text: the

text: ﬂux

text: of

text: carriers

text: of

text: water-

text: the

text: one

text: observed

text: among

text: modern

text: CM

text: and

text: CR

text: falls

text: equivalent

text: material

text: for

text: our

text: results,

text: we

text: consid-

text: on

text: Earth

text: (Zolensky

text: et

text: al.,

text: Therefore,

text: in

text: or-

text: ered

text: also

text: the

text: water

text: delivery

text: eﬃciency

text: discussed

text: der

text: to

text: assess

text: the

text: amount

text: of

text: water-equivalent

text: ma-

text: by

text: Ong

text: et

text: al.

text: In

text: their

text: work,

text: these

text: au-

text: terial

text: that

text: a

text: continuous

text: ﬂux

text: of

text: impactors

text: would

text: thors

text: assumed

text: that

text: about

text: one-third

text: of

text: impacting

text: bring

text: on

text: Vesta,

text: we

text: tried

text: to

text: estimate

text: the

text: frequency

text: asteroids

text: would

text: be

text: C-type

text: (our

text: dark

text: impactors,

text: of

text: CM-,

text: CR-

text: and

text: CI-like

text: projectiles

text: among

text: the

text: i.e.

text: F

text: and

text: that

text: two-thirds

text: of

text: those

text: im-

text: d

text: dark

text: impactors

text: hitting

text: the

text: asteroid.

text: Based

text: on

text: pactors

text: would

text: contain

text: water-equivalent

text: material

text: the

text: observation

text: by

text: Zolensky

text: et

text: al.

text: that

text: (F

text: These

text: authors

text: assumed

text: an

text: average

text: w

text: the

text: CR/CM

text: ratios

text: are

text: similar

text: among

text: meteorite

text: wt%

text: water

text: content,

text: i.e.

text: the

text: same

text: value

text: we

text: falls

text: on

text: Earth

text: and

text: HEDs

text: meteorites,

text: we

text: used

text: the

text: adopted

text: in

text: the

text: previous

text: cases.

text: As

text: these

text: authors

text: ﬂuxes

text: of

text: the

text: falls

text: for

text: the

text: diﬀerent

text: classes

text: of

text: car-

text: point

text: out,

text: their

text: assumptions

text: result

text: in

text: water

text: repre-

text: bonaceous

text: chondrites

text: on

text: Earth

text: as

text: our

text: planetary

text: senting

text: of

text: the

text: total

text: mass

text: ﬂux

text: due

text: to

text: impacts.

text: analogue.

text: From

text: the

text: Meteoritical

text: as

text: As

text: a

text: comparison,

text: in

text: our

text: reference

text: case

text: we

text: have

text: of

text: May

text: we

text: note

text: that

text: CM

text: chondrites

text: repre-

text: F

text: F

text: CM+CR

text: R

text: i.e.

text: our

text: reference

text: w

text: w

text: sent

text: of

text: the

text: falls

text: of

text: carbonaceous

text: chondrites,

text: water

text: delivery

text: eﬃciency

text: is

text: times

text: smaller.

text: CR

text: chondrites

text: represent

text: and

text: CI

text: chondrites

text: Using

text: the

text: value

text: of

text: R

text: and

text: one

text: of

text: the

text: possible

text: w

text: The

text: fraction

text: of

text: the

text: masses

text: of

text: these

text: im-

text: values

text: of

text: F

text: previously

text: discussed,

text: the

text: amount

text: of

text: w

text: pactors

text: that

text: represent

text: water-equivalent

text: material

text: water-equivalent

text: material

text: mw

text: delivered

text: on

text: Vesta

text: i

text: can

text: be

text: assumed,

text: on

text: average,

text: R

text: i.e.

text: by

text: dark

text: impactors

text: of

text: a

text: given

text: size

text: D

text: over

text: a

text: tem-

text: w

text: i

text: wt%

text: (Jarosewich,

text: Robert,

text: poral

text: interval

text: ∆T

text: is

text: then

text: mwi

text: mdiark

text: Fw

text: Rw

text: ##LTLine##