text: 16 FRANC¸OIS GREER of stacks is 2-Cartesian. In particular, B is the coarse space of B. If the family has maximal variation at each singularity then B is smooth. → We apply Theorem 34 to the family ν : S F(Y ) to obtain a Deligne-Mumford stack F(Y ) such that the (primitive) Gauss-Manin system on the smooth locus L extends to a local system on all of F(Y ) with stalk Λ. Let E be the principal L O(Λ)-bundle on F(Y ) of isomorphisms from to the constant sheaf Λ. There is → an equivariant map from E D sending a point in E to the Hodge structure on Λ L obtained by identifying the stalk of with Λ via the isomorphism. This data gives the desired stacky period map → \ F(Y ) [O(Λ) D]. 5. Modularity Statement The work of Kudla-Millson produces a modularity statement for intersection num- bers in a general class of locally symmetric spaces M of orthogonal type. In this section, we summarize3 the material in [15], and adapt it to our situation. \ Let M be the double quotient Γ O(p, l)/K of an orthogonal group on the left by a ⊂ R torsion-free arithmetic subgroup preserving an even unimodular lattice Λ p+l, and on the right by a maximal compact subgroup. This is automatically a mani- fold, since any torsion-free discrete subgroup acts freely on the compact cosets. We can interpret O(p, l)/K as an open subset of the real Grassmannian Gr(p, p + l) ⊂ R consisting of those p-planes Z p+l on which the form is positive deﬁnite. For (cid:101) (cid:104) (cid:105) ⊂ R any negative deﬁnite line v p+l, set { ∈ ⊂ (cid:104) (cid:105)⊥(cid:101)} C(cid:104) (cid:105) := Z O(p, l)/K : Z v v (cid:101) R which is -codimension p. Indeed, the normal bundle to C(cid:104) (cid:105) has ﬁber at Z equal (cid:104) (cid:105) v to Hom(Z, v ). While the image of C(cid:104) (cid:105) in M (cid:101)may be singular, it can always be v resolved if we instead quotient by a ﬁnite index normal subgroup of Γ. Further- more, [15] gives a coherent way of orienting the C(cid:104) (cid:105), so that it makes sense to take v (cid:39) their classes in the Borel-Moore homology group HB−M (M) Hp(M). pl p For any positive integer n, the action of Γ on the lattice vectors in Λ of norm − { } (cid:91) 2n has ﬁnitely many orbits [5]. C(cid:101)hoose orb(cid:101)it representatives v1, . . . , vk , and set k C := C(cid:104) (cid:105). (cid:101) 2n vi (cid:101) i=1 The image of C in the arithmetic quotient M is denoted by C . Locally, C is 2n 2n n a union of smooth (real) codimension p cycles meeting pairwise transversely, one − for each lattice vector of norm 2n orthogonal to Z. We quote the following result from [15]: 3We match the notation of [15] for the most part, but all instances of positive (resp. negative) deﬁniteness are switched. ##LTLine## ##LTLine##

text: FRANC¸OIS

text: GREER

text: of

text: stacks

text: is

text: In

text: particular,

text: B

text: is

text: the

text: coarse

text: space

text: of

text: B.

text: If

text: the

text: family

text: has

text: maximal

text: variation

text: at

text: each

text: singularity

text: then

text: B

text: is

text: smooth.

text: We

text: apply

text: Theorem

text: to

text: the

text: family

text: ν

text: S

text: F(Y

text: to

text: obtain

text: a

text: Deligne-Mumford

text: stack

text: F(Y

text: such

text: that

text: the

text: (primitive)

text: Gauss-Manin

text: system

text: on

text: the

text: smooth

text: locus

text: L

text: extends

text: to

text: a

text: local

text: system

text: on

text: all

text: of

text: F(Y

text: with

text: stalk

text: Λ.

text: Let

text: E

text: be

text: the

text: principal

text: L

text: O(Λ)-bundle

text: on

text: F(Y

text: of

text: isomorphisms

text: from

text: to

text: the

text: constant

text: sheaf

text: Λ.

text: There

text: is

text: an

text: equivariant

text: map

text: from

text: E

text: D

text: sending

text: a

text: point

text: in

text: E

text: to

text: the

text: Hodge

text: structure

text: on

text: Λ

text: L

text: obtained

text: by

text: identifying

text: the

text: stalk

text: of

text: with

text: Λ

text: via

text: the

text: isomorphism.

text: This

text: data

text: gives

text: the

text: desired

text: stacky

text: period

text: map

text: F(Y

text: [O(Λ)

text: D].

text: Modularity

text: Statement

text: The

text: work

text: of

text: Kudla-Millson

text: produces

text: a

text: modularity

text: statement

text: for

text: intersection

text: num-

text: bers

text: in

text: a

text: general

text: class

text: of

text: locally

text: symmetric

text: spaces

text: M

text: of

text: orthogonal

text: type.

text: In

text: this

text: section,

text: we

text: the

text: material

text: in

text: and

text: adapt

text: it

text: to

text: our

text: situation.

text: Let

text: M

text: be

text: the

text: double

text: quotient

text: Γ

text: O(p,

text: l)/K

text: of

text: an

text: orthogonal

text: group

text: on

text: the

text: left

text: by

text: a

text: R

text: torsion-free

text: arithmetic

text: subgroup

text: preserving

text: an

text: even

text: unimodular

text: lattice

text: Λ

text: p+l,

text: and

text: on

text: the

text: right

text: by

text: a

text: maximal

text: compact

text: subgroup.

text: This

text: is

text: automatically

text: a

text: mani-

text: fold,

text: since

text: any

text: torsion-free

text: discrete

text: subgroup

text: acts

text: freely

text: on

text: the

text: compact

text: cosets.

text: We

text: can

text: interpret

text: O(p,

text: l)/K

text: as

text: an

text: open

text: subset

text: of

text: the

text: real

text: Grassmannian

text: Gr(p,

text: p

text: l)

text: R

text: consisting

text: of

text: those

text: p-planes

text: Z

text: p+l

text: on

text: which

text: the

text: form

text: is

text: positive

text: deﬁnite.

text: For

text: R

text: any

text: negative

text: deﬁnite

text: line

text: v

text: p+l,

text: set

text: Z

text: O(p,

text: l)/K

text: Z

text: v

text: v

text: R

text: which

text: is

text: -codimension

text: p.

text: Indeed,

text: the

text: normal

text: bundle

text: to

text: has

text: ﬁber

text: at

text: Z

text: equal

text: v

text: to

text: Hom(Z,

text: v

text: While

text: the

text: image

text: of

text: in

text: M

text: be

text: singular,

text: it

text: can

text: always

text: be

text: v

text: resolved

text: if

text: we

text: instead

text: quotient

text: by

text: a

text: ﬁnite

text: index

text: normal

text: subgroup

text: of

text: Γ.

text: Further-

text: more,

text: gives

text: a

text: coherent

text: way

text: of

text: orienting

text: the

text: so

text: that

text: it

text: makes

text: sense

text: to

text: take

text: v

text: their

text: classes

text: in

text: the

text: Borel-Moore

text: homology

text: group

text: HB−M

text: (M)

text: Hp(M).

text: pl

text: p

text: For

text: any

text: positive

text: integer

text: n,

text: the

text: action

text: of

text: Γ

text: on

text: the

text: lattice

text: vectors

text: in

text: Λ

text: of

text: norm

text: has

text: ﬁnitely

text: many

text: orbits

text: representatives

text: vk

text: and

text: set

text: k

text: C

text: vi

text: The

text: image

text: of

text: C

text: in

text: the

text: arithmetic

text: quotient

text: M

text: is

text: denoted

text: by

text: C

text: Locally,

text: C

text: is

text: n

text: a

text: union

text: of

text: smooth

text: (real)

text: codimension

text: p

text: cycles

text: meeting

text: pairwise

text: transversely,

text: one

text: for

text: each

text: lattice

text: vector

text: of

text: norm

text: orthogonal

text: to

text: Z.

text: We

text: quote

text: the

text: following

text: result

text: from

text: match

text: the

text: notation

text: of

text: for

text: the

text: most

text: part,

text: but

text: all

text: instances

text: of

text: positive

text: (resp.

text: negative)

text: deﬁniteness

text: are

text: switched.

text: ##LTLine##

text: ##LTLine##


text: MODULAR FORMS FROM NOETHER-LEFSCHETZ THEORY 21 → The ﬁrst projection Ω U has ﬁber cut out by µ equations on W, which we µ1 1 wish to beindependent. Setting A(t) = a + a t + . . . and B(t) = b + b t + . . . 0 1 0 1 for t the uniformizer at p, the diﬀerential of the multiplicity condition is given by      3a2 0 0 0 . . . 2b 0 0 . . . 0 0 0 6a a 3a2 0 0 . . . 2b 2b 0 . . . 0 0 1 0 1 0 6a a + 3a2 6a a 3a2 0 . . . 2b 2b 2b . . . 0 0 2 1 0 1 0 2 1 0 . . . The rows are independent unless a = b = 0. In this case, it is easy to check that 0 0 (cid:100) (cid:101) (cid:100) (cid:101) ≤ A(t), resp. B(t), is actually divisible by t µ1/3 , resp. t µ1/2 , when µ1 6. There is an entire irreducible component { ∈ ≥ (cid:100) (cid:101) ≥ (cid:100) (cid:101)} ⊂ Ω = (L, p, A, B) : p L, mult (A) µ /3 , mult (B) µ /2 Ω µ1,II p 1 p 1 whose ﬁber over a pair (L, p) is a linear subspace of W. For µ < 6, Ω is 1 µ1 equidimensional with two irreducible components: ∪ Ω = Ω Ω . µ1,I µ1,II → Now consider the second projection Ω W. It is dominant by a dimension µ1,I − count, so the general ﬁber must have codimension µ 1. Th(cid:88)e case of a general 1 partition is similar; gather only the multiplicities µ greater than 1, and consider j { ∈ | } Ω = (L, p , . . . , p , A, B) : p L, (A2 + B3) = µ p µ 1 l j L j j ⊂ × · · · × × P O × (U G G U) W( m+1, (k)) =: U W. ∈ ∈ Consider the ﬁber at (L, p , . . . , p ) U for distinct points p L. If A and B do 1 l j not simultaneously vanish at any of the points, then the matrix of diﬀerentials can be row reduced to a matrix with blocks of the form 1 | ≤ ≤ ∂n(1, x, x2, x3, . . . ) ; (0 n µ ). n! x x=pj j This is a generalized Vandermonde matrix which has independent rows since the points are distinct. If A and B vanish simultaneously at some p , then in fact they j vanish maximally, which is a linear condition on W of the expected codimension. → Hence, the general ﬁber of Ω U has 2l irreducible components, which collapse µ over the big diagonal. Since the monodromy is trivial, we conclude that Ω itself µ has 2l components. We are only interested in one of them, Ω , which gives the µ,I (cid:3) desired codimension for T (∆). µ ∈ Next, we observe that Type II lines L F(Y ) are always limits of Type I lines ≤ (for k 4), so we can eﬀectively ignore them. Trailing 1’s in the partitions are suppressed for convenience. ⊂ G Lemma 42. Let T (A, B) (1, m + 1) be the locus lines L meeting (A), resp. α,β ∈ ∩ (B), with multiplicity α, resp. β, at a common point p (A) (B). Then we have ⊂ T (A, B) T (∆); 1,2 2 ⊂ T (A, B) T (∆). 2,2 3 These are the only Type II lines which show up after intersecting with F(Y ). ##LTLine##

text: MODULAR

label: paragraph

text: FORMS

text: FROM

text: NOETHER-LEFSCHETZ

text: THEORY

text: The

text: ﬁrst

text: projection

text: Ω

text: U

text: has

text: ﬁber

text: cut

text: out

text: by

text: µ

text: equations

text: on

text: W,

text: which

text: we

text: µ1

text: wish

text: to

text: beindependent.

text: Setting

text: A(t)

text: a

text: a

text: t

text: and

text: B(t)

text: b

text: b

text: t

text: for

text: t

text: the

text: uniformizer

text: at

text: p,

text: the

text: diﬀerential

text: of

text: the

text: multiplicity

text: condition

text: is

text: given

text: by

text: 0

text: a

text: a

text: a

text: The

text: rows

text: are

text: independent

text: unless

text: a

text: b

text: In

text: this

text: case,

text: it

text: is

text: easy

text: to

text: check

text: that

text: A(t),

text: resp.

text: B(t),

text: is

text: actually

text: divisible

text: by

text: t

text: resp.

text: t

text: when

text: µ1

text: There

text: is

text: an

text: entire

text: irreducible

text: component

text: Ω

text: (L,

text: p,

text: A,

text: B)

text: p

text: L,

text: mult

text: (A)

text: µ

text: mult

text: (B)

text: µ

text: Ω

text: p

text: p

text: whose

text: ﬁber

text: over

text: a

text: pair

text: (L,

text: p)

text: is

text: a

text: linear

text: subspace

text: of

text: W.

text: For

text: µ

text: Ω

text: is

text: µ1

text: equidimensional

text: with

text: two

text: irreducible

text: components:

text: Ω

text: Ω

text: Ω

text: Now

text: consider

text: the

text: second

text: projection

text: Ω

text: W.

text: It

text: is

text: dominant

text: by

text: a

text: dimension

text: count,

text: so

text: the

text: general

text: ﬁber

text: must

text: have

text: codimension

text: µ

text: case

text: of

text: a

text: general

text: partition

text: is

text: similar;

text: gather

text: only

text: the

text: multiplicities

text: µ

text: greater

text: than

text: and

text: consider

text: j

text: Ω

text: (L,

text: p

text: p

text: A,

text: B)

text: p

text: L,

text: µ

text: p

text: µ

text: l

text: j

text: L

text: j

text: j

text: P

text: O

text: (U

text: G

text: G

text: U)

text: W(

text: (k))

text: U

text: W.

text: Consider

text: the

text: ﬁber

text: at

text: (L,

text: p

text: p

text: U

text: for

text: distinct

text: points

text: p

text: L.

text: If

text: A

text: and

text: B

text: do

text: l

text: j

text: not

text: simultaneously

text: vanish

text: at

text: any

text: of

text: the

text: points,

text: then

text: the

text: matrix

text: of

text: diﬀerentials

text: can

text: be

text: row

text: reduced

text: to

text: a

text: matrix

text: with

text: blocks

text: of

text: the

text: form

text: x,

text: n

text: µ

text: n!

text: x

text: x=pj

text: j

text: This

text: is

text: a

text: generalized

text: Vandermonde

text: matrix

text: which

text: has

text: independent

text: rows

text: since

text: the

text: points

text: are

text: distinct.

text: If

text: A

text: and

text: B

text: vanish

text: simultaneously

text: at

text: some

text: p

text: then

text: in

text: fact

text: they

text: j

text: vanish

text: maximally,

text: which

text: is

text: a

text: linear

text: condition

text: on

text: W

text: of

text: the

text: expected

text: codimension.

text: Hence,

text: the

text: general

text: ﬁber

text: of

text: Ω

text: U

text: has

text: irreducible

text: components,

text: which

text: collapse

text: µ

text: over

text: the

text: big

text: diagonal.

text: Since

text: the

text: monodromy

text: is

text: trivial,

text: we

text: conclude

text: that

text: Ω

text: itself

text: µ

text: has

text: components.

text: We

text: are

text: only

text: interested

text: in

text: one

text: of

text: them,

text: Ω

text: which

text: gives

text: the

text: µ,I

text: desired

text: codimension

text: for

text: T

text: µ

text: Next,

text: we

text: observe

text: that

text: Type

text: II

text: lines

text: L

text: F(Y

text: are

text: always

text: limits

text: of

text: Type

text: I

text: lines

text: (for

text: k

text: so

text: we

text: can

text: eﬀectively

text: ignore

text: them.

text: Trailing

text: in

text: the

text: partitions

text: are

text: suppressed

text: for

text: convenience.

text: G

text: Lemma

text: Let

text: T

text: (A,

text: B)

text: m

text: be

text: the

text: locus

text: lines

text: L

text: meeting

text: (A),

text: resp.

text: α,β

text: (B),

text: with

text: multiplicity

text: α,

text: resp.

text: β,

text: at

text: a

text: common

text: point

text: p

text: (A)

text: (B).

text: Then

text: we

text: have

text: T

text: (A,

text: B)

text: T

text: T

text: (A,

text: B)

text: T

text: These

text: are

text: the

text: only

text: Type

text: II

text: lines

text: which

text: show

text: up

text: after

text: intersecting

text: with

text: F(Y

text: ##LTLine##


text: 24 FRANC¸OIS GREER which agrees with the cuspidal correction from Proposition . Finally, we use the (cid:48)(cid:48) genus formula on R : (cid:101) − (cid:48)(cid:48) · (cid:48) (cid:48) 2g 2 = (K + R ) R ; (cid:48)(cid:48) −S − − δ(B ) = 12k(9k 1)(6k 1)(4k 1). (cid:3) Subtracting the ﬂex line count from Proposition leaves the desired number. In the sequel, we will use the notation · t := [T (∆)] F(Y ) µ µ (cid:88) when this intersection is 0 dimensional, that is l − − (µ 1) = k 1 j j=1 (cid:89) By the results of Section , the stack F(Y ) will have isotropy group l S µj j=1 at these isolated points. 7. Counting Curves We begin by discussing the case k = 1, which is trivial because the period domain is a point. Since F(Y ) is 0-dimensional, let N = #F(Y ), and we may assume that ∈ m − for each [L] F(Y ), the rational elliptic surface S = π 1(L) is smooth. Every class (cid:39) (cid:88) in the polarized N´eron-Severi lattice NS (S) E is the orthogonal projection of 0 8 a section curve. The degree shifts from Lemmas 22 and 24 match, so we have − r (n)qn = N θ (q) N ≥ X m E8 m n 1 and θ (q) is the modular form of weight 4, as desired. E8 ≤ ≤ → For 2 k 4, recall the family of elliptic surfaces ν : S F(Y ) deﬁned by the diagram (cid:47)(cid:47) S X . ν (cid:48) π π (cid:124)(cid:124) (cid:15)(cid:15) (cid:15)(cid:15) (cid:111)(cid:111) (cid:47)(cid:47) F(Y ) U Y. → \ → The period map j : F(Y ) O(Λ) D lifts to an immersion of stacks j : F(Y ) \ [O(Λ) D] of Deligne-Mumford type. Theorem 38 applied to α = j∗[F(Y )] yields − a classical modular form ϕ(q) of weight 6k 2 and full level. Our ﬁrst task is to determine this modular form. We start by computing the constant term as the top Chern class of the dual Hodge bundle: → Deﬁnition 46. The Hodge bundle of t(cid:0)he family(cid:1)ν : S F(Y ) is deﬁned as the pushforward of the relative dualizing sheaf: λ := ν∗ ωS/F(Y ) . Proposition 47. Let S be the tautological rank 2 bundle on F(Y ). Then (cid:39) − ∨ ⊗ O λ Symk 2(S ) (σ ) F(Y ) 1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: FRANC¸OIS

text: GREER

text: which

text: agrees

text: with

text: the

text: cuspidal

text: correction

text: from

text: Proposition

text: Finally,

text: we

text: use

text: the

text: genus

text: formula

text: on

text: R

text: (K

text: R

text: R

text: −S

text: δ(B

text: Subtracting

text: the

text: ﬂex

text: line

text: count

text: from

text: Proposition

text: leaves

text: the

text: desired

text: number.

text: In

text: the

text: sequel,

text: we

text: will

text: use

text: the

text: notation

text: t

text: [T

text: F(Y

text: µ

text: µ

text: when

text: this

text: intersection

text: is

text: dimensional,

text: that

text: is

text: l

text: (µ

text: k

text: j

text: By

text: the

text: results

text: of

text: Section

text: the

text: stack

text: F(Y

text: will

text: have

text: isotropy

text: group

text: l

text: S

text: µj

text: at

text: these

text: isolated

text: points.

text: Counting

text: Curves

text: We

text: begin

text: by

text: discussing

text: the

text: case

text: k

text: which

text: is

text: trivial

text: because

text: the

text: period

text: domain

text: is

text: a

text: point.

text: Since

text: F(Y

text: is

text: let

text: N

text: #F(Y

text: and

text: we

text: may

text: assume

text: that

text: m

text: for

text: each

text: [L]

text: F(Y

text: the

text: rational

text: elliptic

text: surface

text: S

text: π

text: is

text: smooth.

text: Every

text: class

text: in

text: the

text: polarized

text: N´eron-Severi

text: lattice

text: NS

text: (S)

text: E

text: is

text: the

text: orthogonal

text: projection

text: of

text: a

text: section

text: curve.

text: The

text: degree

text: shifts

text: from

text: Lemmas

text: and

text: match,

text: so

text: we

text: have

text: r

text: (n)qn

text: N

text: θ

text: (q)

text: N

text: X

text: m

text: m

text: n

text: and

text: θ

text: (q)

text: is

text: the

text: modular

text: form

text: of

text: weight

text: as

text: desired.

text: For

text: k

text: recall

text: the

text: family

text: of

text: elliptic

text: surfaces

text: ν

text: S

text: F(Y

text: deﬁned

text: by

text: the

text: diagram

text: S

text: X

text: ν

text: π

text: π

text: F(Y

text: U

text: Y.

text: The

text: period

text: map

text: j

text: F(Y

text: O(Λ)

text: D

text: lifts

text: to

text: an

text: immersion

text: of

text: stacks

text: j

text: F(Y

text: [O(Λ)

text: D]

text: of

text: Deligne-Mumford

text: type.

text: Theorem

text: applied

text: to

text: α

text: j∗[F(Y

text: yields

text: a

text: classical

text: modular

text: form

text: ϕ(q)

text: of

text: weight

text: and

text: full

text: level.

text: Our

text: ﬁrst

text: task

text: is

text: to

text: determine

text: this

text: modular

text: form.

text: We

text: start

text: by

text: computing

text: the

text: constant

text: term

text: as

text: the

text: top

text: Chern

text: class

text: of

text: the

text: dual

text: Hodge

text: bundle:

text: Deﬁnition

text: The

text: Hodge

text: bundle

text: of

text: S

text: F(Y

text: is

text: deﬁned

text: as

text: the

text: pushforward

text: of

text: the

text: relative

text: dualizing

text: sheaf:

text: λ

text: ν∗

text: ωS/F(Y

text: Proposition

text: Let

text: S

text: be

text: the

text: tautological

text: rank

text: bundle

text: on

text: F(Y

text: Then

text: O

text: λ

text: Symk

text: (σ

text: F(Y

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4 FRANC¸OIS GREER (cid:88) Recall that for a lattice A, the associated theta series is given by θ (q) = q(v,v)/2, A ∈ v A and we assign the weight ρ to the series θ for the root lattice A . Aρ ρ In short, the curve counts r (n) are controlled by a ﬁnite amount of data, since − Z Q X Q Mod(6k 2, SL ( )) and [θ , θ , . . . ] are ﬁnite dimensional -vector spaces. 2 A1 A2 <k ≤ The series can be explicitly computed when k 3. We record a few examples in Section to illustrate the scope of applicability. ≤ Remark 7. We expect that Theorem 6 can be extended to k 8, but the statement is less tidy and involves the root systems D , E , and E . 4 6 7 The argument proceeds roughly as follows. The curves C that we wish to count ⊂ have the property that π(C) Y is a line. In other words, they can be viewed as sections of the elliptic ﬁbration − → (cid:39) P π 1(π(C)) π(C) 1. ⊂ As we vary over lines L Y , we obtain a family of elliptic surfaces over the Fano variety of lines in Y : → ν : S F(Y ). ∈ − We would like to count points [L] F(Y ) such that S = π 1(L) has a non- [L] identity section, i.e. a non-trivial Mordell-Weil group. The Shioda-Tate sequence expresses the Mordell-Weil group of an elliptic surface in terms of its N´eron-Severi lattice and the sublattice V (S) spanned by vertical classes and the identity section: → → → P → 0 V (S) NS(S) MW(S/ 1) 0. The period domain for a given class of elliptic surfaces is related to a locally sym- metric space, whence the modular form ϕ(q), which counts surfaces with jumping Picard rank. To obtain the counts r (n) we subtract contributions Θ(q) from sur- X faces with jumping V (S), which are precisely those with A singularities. Hence, ρ the terms in the diﬀerence formula of Theorem 6 are matched to the groups in the sequence. The paper is organized as follows. In Section , we review the relevant theory of elliptic ﬁbrations and set up the tools for proving transversality of intersections in moduli. In Section , we review the theory of period domains and lattices in the co- homology of elliptic surfaces. Section is devoted to the deformation and resolution ##LTFigure##

text: FRANC¸OIS

text: GREER

text: Recall

text: that

text: for

text: a

text: lattice

text: A,

text: the

text: associated

text: theta

text: series

text: is

text: given

text: by

text: θ

text: (q)

text: A

text: v

text: A

text: and

text: we

text: assign

text: the

text: weight

text: ρ

text: to

text: the

text: series

text: θ

text: for

text: the

text: root

text: lattice

text: A

text: Aρ

text: ρ

text: In

text: short,

text: the

text: curve

text: counts

text: r

text: (n)

text: are

text: controlled

text: by

text: a

text: ﬁnite

text: amount

text: of

text: data,

text: since

text: Z

text: Q

text: X

text: Q

text: SL

text: and

text: [θ

text: θ

text: are

text: ﬁnite

text: dimensional

text: -vector

text: spaces.

text: <k

text: The

text: series

text: can

text: be

text: explicitly

text: computed

text: when

text: k

text: We

text: record

text: a

text: few

text: examples

text: in

text: Section

text: to

text: illustrate

text: the

text: scope

text: of

text: applicability.

text: Remark

text: We

text: expect

text: that

text: Theorem

text: can

text: be

text: extended

text: to

text: k

text: but

text: the

text: statement

text: is

text: less

text: tidy

text: and

text: involves

text: the

text: root

text: systems

text: D

text: E

text: and

text: E

text: The

text: argument

text: proceeds

text: roughly

text: as

text: follows.

text: The

text: curves

text: C

text: that

text: we

text: wish

text: to

text: count

text: have

text: the

text: property

text: that

text: π(C)

text: Y

text: is

text: a

text: line.

text: In

text: other

text: words,

text: they

text: can

text: be

text: viewed

text: as

text: sections

text: of

text: the

text: elliptic

text: ﬁbration

text: P

text: π

text: π(C)

text: As

text: we

text: vary

text: over

text: lines

text: L

text: Y

text: we

text: obtain

text: a

text: family

text: of

text: elliptic

text: surfaces

text: over

text: the

text: Fano

text: variety

text: of

text: lines

text: in

text: Y

text: ν

text: S

text: F(Y

text: We

text: would

text: like

text: to

text: count

text: points

text: [L]

text: F(Y

text: such

text: that

text: S

text: π

text: has

text: a

text: non-

text: [L]

text: identity

text: section,

text: i.e.

text: a

text: non-trivial

text: Mordell-Weil

text: group.

text: The

text: Shioda-Tate

text: sequence

text: expresses

text: the

text: Mordell-Weil

text: group

text: of

text: an

text: elliptic

text: surface

text: in

text: terms

text: of

text: its

text: N´eron-Severi

text: lattice

text: and

text: the

text: sublattice

text: V

text: (S)

text: spanned

text: by

text: vertical

text: classes

text: and

text: the

text: identity

text: section:

text: P

text: V

text: (S)

text: NS(S)

text: MW(S/

text: The

text: period

text: domain

text: for

text: a

text: given

text: class

text: of

text: elliptic

text: surfaces

text: is

text: related

text: to

text: a

text: locally

text: sym-

text: metric

text: space,

text: whence

text: the

text: modular

text: form

text: ϕ(q),

text: which

text: counts

text: surfaces

text: with

text: jumping

text: Picard

text: rank.

text: To

text: obtain

text: the

text: counts

text: r

text: (n)

text: we

text: subtract

text: contributions

text: Θ(q)

text: from

text: sur-

text: X

text: faces

text: with

text: jumping

text: V

text: (S),

text: which

text: are

text: precisely

text: those

text: with

text: A

text: singularities.

text: Hence,

text: ρ

text: the

text: terms

text: in

text: the

text: diﬀerence

text: formula

text: of

text: Theorem

text: are

text: matched

text: to

text: the

text: groups

text: in

text: the

text: sequence.

text: The

text: paper

text: is

text: organized

text: as

text: follows.

text: In

text: Section

text: we

text: review

text: the

text: relevant

text: theory

text: of

text: elliptic

text: ﬁbrations

text: and

text: set

text: up

text: the

text: tools

text: for

text: proving

text: transversality

text: of

text: intersections

text: in

text: moduli.

text: In

text: Section

text: we

text: review

text: the

text: theory

text: of

text: period

text: domains

text: and

text: lattices

text: in

text: the

text: co-

text: homology

text: of

text: elliptic

text: surfaces.

text: Section

text: is

text: devoted

text: to

text: the

text: deformation

text: and

text: resolution

text: ##LTFigure##


text: 2 MARIA ANGELICA CUETO AND HANNAH MARKWIG { } forget hyperelliptic (cid:47)(cid:47) (cid:111)(cid:111) planar sm. genus 2 hyperelliptic curves M (K) M (K) 2 cover 0,6 § § ϕ § trop trop § trop (cid:15)(cid:15) (cid:15)(cid:15) (cid:120)(cid:120) (cid:15)(cid:15) { } forget trop. hyperell. faithful tropical genus 2 hyperell. curves § (cid:47)(cid:47) Mtrop (cid:111)(cid:111) § Mtrop 2 cover 0,6 Figure 1. Three ways to represent genus two curves, their relations, and their tropical analogues. moduli space of abstract genus two tropical curves Mtrop is obtained as the image of 2 M (K) under the tropicalization map [1, Theorem 1.2.1]. It has the structure of a stacky 2 fan with seven cones, each labeled by a type and isomorphic to an orthant of dimension equal to the number of edges on the skeleton [1, 16, 18]. We dicuss this space in more detail in Section . The tropical moduli space Mtrop of rational tropical curves with six marked points is 0,6 the space of phylogenetic trees on six leaves of Billera-Holmes-Vogtmann [7]. It is realized as the image of M (K) under the vertical tropicalization map in Figure 1, i.e., by taking 0,6 coordinatewise negative valuations of all K-points of M embedded in the toric variety deﬁned by the pointed fan Mtrop ⊂ R9. This map an0d,6the combinatorial structure of 0,6 Mtrop are also discussed in Section . 0,6 As in the algebraic case, abstract genus two tropical curves are hyperelliptic: they admit a tropical hyperelliptic cover of a metric tree with six markings, given by a 2-to-1 harmonic map branched at all six legs of the tree [3, 18]. We review this construction in Section . The tropical covers turn the right square of Figure 1 into a commuting diagram, but the assignment is not explicit: it requires prior knowledge of each Berkovich skeleta. We bypass this diﬃculty by factoring the right square of the diagram through the map ϕ. The ∈ ∗ assignment depends on the valuations of the points α , . . . , α K and their diﬀerences: 1 6 − − − (1.2) ω := val(α ) for i = 1, . . . , 6, and d := val(α α ) for i < j, if ω = ω . i i ij i j i j Here is our ﬁrst main result, which we discuss in Section : Theorem 1.1. Each point in Mtrop together with an explicit harmonic 2-to-1 map to a 2 metric tree in Mtrop is determined by the ordering of the quantities ω and d (see Table 1). 0,6 i ij For example, the two maximal cells in Mtrop correspond to the orders ω <ω <ω <ω < 2 ≤ 1 2 3 4 ω <ω (the dumbbell graph (I)) and ω <ω <ω ω <ω <ω with d >ω (the theta 5 6 1 2 3 4 5 6 34 3 graph (II)). They are realized as 2-to-1 harmonic covers of the caterpillar and snowﬂake trees as shown in Figure 2. Similar results were obtained earlier by Ren-Sam-Sturmfels [44, Table 3] but with very diﬀerent methods. TP Our proof of Theorem 1.1 is sketched in the right of Figure 2. Starting from 1, tropical TP modiﬁcations of 1 at the locations of the points ω dictated by the quantities d allow i ij us to construct the target metric trees. The source curve and the map are determined by the tropical Riemann-Hurwitz formula [13]. Proposition 5.2 provides a list of seven regions in M (K) that surject onto Mtrop. Algorithms 1 and 2 take six arbitrary points ∗ 0,6 2 P in (K )6 and return a linear change of coordinates of 1 that sends these six points to one of these seven witness regions. The same techniques will lead to a natural extension ≥ of Theorem 1.1 to the tropical hyperelliptic locus in Mtrop for any g 2. g ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: MARIA

text: ANGELICA

text: CUETO

text: AND

text: HANNAH

text: MARKWIG

text: forget

text: hyperelliptic

font: default

text: planar

text: sm.

text: genus

text: hyperelliptic

text: curves

text: M

text: (K)

text: M

text: (K)

text: cover

text: ϕ

text: trop

text: trop

text: trop

text: forget

text: trop.

text: hyperell.

text: faithful

text: tropical

text: genus

text: hyperell.

text: curves

text: Mtrop

text: Mtrop

text: cover

text: Figure

text: Three

label: caption

text: ways

text: to

text: represent

text: genus

text: two

text: curves,

text: their

text: relations,

text: and

text: their

text: tropical

text: analogues.

text: moduli

text: space

text: of

text: abstract

text: genus

text: two

text: tropical

text: curves

text: Mtrop

text: is

text: obtained

text: as

text: the

text: image

text: of

text: M

text: (K)

text: under

text: the

text: tropicalization

text: map

text: Theorem

text: It

text: has

text: the

text: structure

text: of

text: a

text: stacky

text: fan

text: with

text: seven

text: cones,

text: each

text: labeled

text: by

text: a

text: type

text: and

text: isomorphic

text: to

text: an

text: orthant

text: of

text: dimension

text: equal

text: to

text: the

text: number

text: of

text: edges

text: on

text: the

text: skeleton

text: We

text: dicuss

text: this

text: space

text: in

text: more

text: detail

text: in

text: Section

text: The

text: tropical

text: moduli

text: space

text: Mtrop

text: of

text: rational

text: tropical

text: curves

text: with

text: six

text: marked

text: points

text: is

text: the

text: space

text: of

text: phylogenetic

text: trees

text: on

text: six

text: leaves

text: of

text: Billera-Holmes-Vogtmann

text: It

text: is

text: realized

text: as

text: the

text: image

text: of

text: M

text: (K)

text: under

text: the

text: vertical

text: tropicalization

text: map

text: in

text: Figure

text: i.e.,

text: by

text: taking

text: coordinatewise

text: negative

text: valuations

text: of

text: all

text: K-points

text: of

text: M

text: embedded

text: in

text: the

text: toric

text: variety

text: deﬁned

text: by

text: the

text: pointed

text: fan

text: Mtrop

text: This

text: map

text: combinatorial

text: structure

text: of

text: Mtrop

text: are

text: also

text: discussed

text: in

text: Section

text: As

text: in

text: the

text: algebraic

text: case,

text: abstract

text: genus

text: two

text: tropical

text: curves

text: are

text: hyperelliptic:

text: they

text: admit

text: a

text: tropical

text: hyperelliptic

text: cover

text: of

text: a

text: metric

text: tree

text: with

text: six

text: markings,

text: given

text: by

text: a

text: harmonic

text: map

text: branched

text: at

text: all

text: six

text: legs

text: of

text: the

text: tree

text: We

text: review

text: this

text: construction

text: in

text: Section

text: The

text: tropical

text: covers

text: turn

text: the

text: right

text: square

text: of

text: Figure

text: into

text: a

text: commuting

text: diagram,

text: but

text: the

text: assignment

text: is

text: not

text: explicit:

text: it

text: requires

text: prior

text: knowledge

text: of

text: each

text: Berkovich

text: skeleta.

text: We

text: bypass

text: this

text: diﬃculty

text: by

text: factoring

text: the

text: right

text: square

text: of

text: the

text: diagram

text: through

text: the

text: map

text: ϕ.

text: The

text: assignment

text: depends

text: on

text: the

text: valuations

text: of

text: the

text: points

text: α

text: α

text: K

text: and

text: their

text: diﬀerences:

text: ω

text: val(α

text: for

text: i

text: and

text: d

text: val(α

text: α

text: for

text: i

text: j,

text: if

text: ω

text: ω

text: i

text: i

text: ij

text: i

text: j

text: i

text: j

text: Here

text: is

text: our

text: ﬁrst

text: main

text: result,

text: which

text: we

text: discuss

text: in

text: Section

text: Theorem

text: Each

text: point

text: in

text: Mtrop

text: together

text: with

text: an

text: explicit

text: harmonic

text: map

text: to

text: a

text: metric

text: tree

text: in

text: Mtrop

text: is

text: determined

text: by

text: the

text: ordering

text: of

text: the

text: quantities

text: ω

text: and

text: d

text: (see

text: Table

text: i

text: ij

text: For

text: example,

text: the

text: two

text: maximal

text: cells

text: in

text: Mtrop

text: correspond

text: to

text: the

text: orders

text: ω

text: <ω

text: <ω

text: <ω

text: ω

text: <ω

text: (the

text: dumbbell

text: graph

text: (I))

text: and

text: ω

text: <ω

text: <ω

text: ω

text: <ω

text: <ω

text: with

text: d

text: >ω

text: (the

text: theta

text: graph

text: (II)).

text: They

text: are

text: realized

text: as

text: harmonic

text: covers

text: of

text: the

text: caterpillar

text: and

text: snowﬂake

text: trees

text: as

text: shown

text: in

text: Figure

text: Similar

text: results

text: were

text: obtained

text: earlier

text: by

text: Ren-Sam-Sturmfels

text: Table

text: but

text: with

text: very

text: diﬀerent

text: methods.

text: TP

text: Our

text: proof

text: of

text: Theorem

text: is

text: sketched

text: in

text: the

text: right

text: of

text: Figure

text: Starting

text: from

text: tropical

text: TP

text: modiﬁcations

text: of

text: at

text: the

text: locations

text: of

text: the

text: points

text: ω

text: dictated

text: by

text: the

text: quantities

text: d

text: allow

text: i

text: ij

text: us

text: to

text: construct

text: the

text: target

text: metric

text: trees.

text: The

text: source

text: curve

text: and

text: the

text: map

text: are

text: determined

text: by

text: the

text: tropical

text: Riemann-Hurwitz

text: formula

text: Proposition

text: provides

text: a

text: list

text: of

text: seven

text: regions

text: in

text: M

text: (K)

text: that

text: surject

text: onto

text: Mtrop.

text: Algorithms

text: and

text: take

text: six

text: arbitrary

text: points

text: P

text: in

text: (K

text: and

text: return

text: a

text: linear

text: change

text: of

text: coordinates

text: of

text: that

text: sends

text: these

text: six

text: points

text: to

text: one

text: of

text: these

text: seven

text: witness

text: regions.

text: The

text: same

text: techniques

text: will

text: lead

text: to

text: a

text: natural

text: extension

text: of

text: Theorem

text: to

text: the

text: tropical

text: hyperelliptic

text: locus

text: in

text: Mtrop

text: for

text: any

text: g

text: g

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 12 MARIA ANGELICA CUETO AND HANNAH MARKWIG can label each tree by the type of the genus two metric graph Γ covering it. The edge length formulas on Γ indicated on the last column of Table 1 are obtained directly from the metric structure on each tree using Remark 4.8. In what remains, we discuss the second column of the table. The combinatorial type of −(cid:39)→ G ∗ ⊂ ∗ ∗ each tree is determined by the isomorphism Φ: M r (2, 6)/(K )6 (K )15/(K )6 0,6 0 from (2.2) and the four-point conditions (i.e., the tropical 3-term Plu¨cker relations) on − ∈ R R val(Φ(α)) 15/ 6. We order the coordinates of Φ(α) lexicographically. Type (I): We claim Trop X is a trivalent caterpillar tree on six leaves with internal edge − − − − − lengths ω ω , ω ω and ω ω . Indeed, since val(α α ) = ω for i < j we have 3 2 4 3 5 4 i j j − ∈ G R val(Φ(α)):= (ω , ω , ω , ω , ω , ω , ω , ω , ω , ω , ω , ω , ω , ω , ω ) Trop r (2, 6)/ 6. 2 3 4 5 6 3 4 5 6 4 5 6 5 6 6 0 P The four-point condition implies that the corresponding line in 5 is a trivalent caterpillar tree. Linear algebra recovers the expected lengths on its three bounded edges. Note that the lengths assigned to the six legs in the second column of Table 1 play no role here: R the associated distance vector in 15 is in the same class modulo the lineality space in G Trop r (2, 6). The claim follows. 0 Type (II): By construction, Φ(α) has negative valuation vector − ∈ G R val(Φ(α)):= (ω , ω , ω , ω , ω , ω , ω , ω , ω , d , ω , ω , ω , ω , ω ) Trop r (2, 6)/ 6. 2 3 4 5 6 3 4 5 6 34 5 6 5 6 6 0 P The four–point conditions imply that the tropical line in 5 is a snowﬂake tree with internal − − − edges ω ω , ω ω and ω d , as indicated on the second column of the table. 3 2 5 3 3 34 Types (III) through (VII): The tropicalization induced by the Plu¨cker embedding shows that the metric trees on these lower-dimensional cells of Mtrop are obtained by 0,6 specializing the trees for Type (I) or Type (II): both the combinatorial type and the metric are obtained by coarsening either the caterpillar or the snowﬂake trees. The edge (cid:3) length formulas match those given in Table 1. In the remaining of this section we discuss why these seven regions Ω(i) suﬃce to clas- sify smooth genus two tropical curves. Indeed, Algorithms 1 and 2 describe an explicit ∗ combinatorial procedure that takes six distinct points α , . . . , α in K and provides lin- 1 6 P ear changes of coordinates of 1 producing a tuple of points in one of the sets Ω(i), after iteratively combining two steps: (A) Separate points: We take a coordinate ω of ω and two points α and α − − k i j of valuation ω where val(α α ) is maximal, and make a linear change of k i ∈j ∗ (cid:48) ∈ ∗ − (cid:48) coordinates that turns the tuple α (K )6 into α (K )6, where val(α ) is the (cid:48) i unique smallest element of ω . The method is described in Lemma 5.3. P (B) Turn around: We change coordinates from one open aﬃne chart of 1 to an- − (cid:48) other by replacing x by 1/x. As a result, val(α ) = val(α ) and the relative order i i (cid:48) of the valuations on the tuple α is reversed on the new tuple α . Lemma 5.3. [Separating points] Consider a repeated coordinate ω of ω, and write { − (cid:54) } ≥ − (cid:101) β = max val(α α ): ω = ω = ω for m = l ω. m l m l − − ∈ Fix two indices i, j with ω =ω =ω and β = val(α α ). If in(α α ) = ζ K for some i ∈ j ∗ i j − i− j ζ with val(ζ) = 0, choose γ val(K ) with β < γ < val(α α ζtβ). Then, the linear P → P i j change of coordinates ψ : 1 1 deﬁned locally by − − − (5.3) ψ(x) = x α ζtβ tγ j ∈ ∗ (cid:48) ∈ ∗ turns the tuple α (K )6 into α (K )6, where their coordinatewise negative valua- (cid:48) tions ω and ω satisfy the following properties: ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: MARIA

text: ANGELICA

text: CUETO

text: AND

text: HANNAH

text: MARKWIG

text: can

text: label

text: each

text: tree

text: by

text: the

text: type

text: of

text: the

text: genus

text: two

text: metric

text: graph

text: Γ

text: covering

text: it.

text: The

text: edge

text: length

text: formulas

text: on

text: Γ

text: indicated

text: on

text: the

text: last

text: column

text: of

text: Table

text: are

text: obtained

text: directly

text: from

text: the

text: metric

text: structure

text: on

text: each

text: tree

text: using

text: Remark

text: In

text: what

text: remains,

text: we

text: discuss

text: the

text: second

text: column

text: of

text: the

text: table.

text: The

text: combinatorial

text: type

text: of

text: G

text: each

text: tree

text: is

text: determined

text: by

text: the

text: isomorphism

text: Φ:

text: M

text: r

text: (K

text: from

text: and

text: the

text: four-point

text: conditions

text: (i.e.,

text: the

text: tropical

text: Plu¨cker

text: relations)

text: on

text: R

text: R

text: val(Φ(α))

text: We

text: order

text: the

text: coordinates

text: of

text: Φ(α)

text: lexicographically.

text: Type

text: (I):

text: We

text: claim

text: Trop

text: X

text: is

text: a

text: trivalent

text: caterpillar

text: tree

text: on

text: six

text: leaves

text: with

text: internal

text: edge

text: lengths

text: ω

text: ω

text: ω

text: ω

text: and

text: ω

text: ω

text: Indeed,

text: since

text: val(α

text: α

text: ω

text: for

text: i

text: j

text: we

text: have

text: i

text: j

text: j

text: G

text: R

text: val(Φ(α)):=

text: (ω

text: ω

text: ω

text: ω

text: ω

text: ω

text: ω

text: ω

text: ω

text: ω

text: ω

text: ω

text: ω

text: ω

text: ω

text: Trop

text: r

text: P

text: The

text: four-point

text: condition

text: implies

text: that

text: the

text: corresponding

text: line

text: in

text: is

text: a

text: trivalent

text: caterpillar

text: tree.

text: Linear

text: algebra

text: recovers

text: the

text: expected

text: lengths

text: on

text: its

text: three

text: bounded

text: edges.

text: Note

text: that

text: the

text: lengths

text: assigned

text: to

text: the

text: six

text: legs

text: in

text: the

text: second

text: column

text: of

text: Table

text: play

text: no

text: role

text: here:

text: R

text: the

text: associated

text: distance

text: vector

text: in

text: is

text: in

text: the

text: same

text: class

text: modulo

text: the

text: lineality

text: space

text: in

text: G

text: Trop

text: r

text: The

text: claim

text: follows.

text: Type

text: (II):

text: By

text: construction,

text: Φ(α)

text: has

text: negative

text: valuation

text: vector

text: G

text: R

text: val(Φ(α)):=

text: (ω

text: ω

text: ω

text: ω

text: ω

text: ω

text: ω

text: ω

text: ω

text: d

text: ω

text: ω

text: ω

text: ω

text: ω

text: Trop

text: r

text: P

text: The

text: four–point

text: conditions

text: imply

text: that

text: the

text: tropical

text: line

text: in

text: is

text: a

text: snowﬂake

text: tree

text: with

text: internal

text: edges

text: ω

text: ω

text: ω

text: ω

text: and

text: ω

text: d

text: as

text: indicated

text: on

text: the

text: second

text: column

text: of

text: the

text: table.

text: Types

text: (III)

text: through

text: (VII):

text: The

text: tropicalization

text: induced

text: by

text: the

text: Plu¨cker

text: embedding

text: shows

text: that

text: the

text: metric

text: trees

text: on

text: these

text: lower-dimensional

text: cells

text: of

text: Mtrop

text: are

text: obtained

text: by

text: specializing

text: the

text: trees

text: for

text: Type

text: (I)

text: or

text: Type

text: (II):

text: both

text: the

text: combinatorial

text: type

text: and

text: the

text: metric

text: are

text: obtained

text: by

text: coarsening

text: either

text: the

text: caterpillar

text: or

text: the

text: snowﬂake

text: trees.

text: The

text: edge

text: length

text: formulas

text: match

text: those

text: given

text: in

text: Table

text: In

text: the

text: remaining

text: of

text: this

text: section

text: we

text: discuss

text: why

text: these

text: seven

text: regions

text: Ω(i)

text: suﬃce

text: to

text: clas-

text: sify

text: smooth

text: genus

text: two

text: tropical

text: curves.

text: Indeed,

text: Algorithms

text: and

text: describe

text: an

text: explicit

text: combinatorial

text: procedure

text: that

text: takes

text: six

text: distinct

text: points

text: α

text: α

text: in

text: K

text: and

text: provides

text: lin-

text: P

text: ear

text: changes

text: of

text: coordinates

text: of

text: producing

text: a

text: tuple

text: of

text: points

text: in

text: one

text: of

text: the

text: sets

text: Ω(i),

text: after

text: iteratively

text: combining

text: two

text: steps:

text: (A)

text: Separate

text: points:

text: We

text: take

text: a

text: coordinate

text: ω

text: of

text: ω

text: and

text: two

text: points

text: α

text: and

text: α

text: k

text: i

text: j

text: of

text: valuation

text: ω

text: where

text: val(α

text: α

text: is

text: maximal,

text: and

text: make

text: a

text: linear

text: change

text: of

text: k

text: i

text: ∈j

text: coordinates

text: that

text: turns

text: the

text: tuple

text: α

text: (K

text: into

text: α

text: (K

text: where

text: val(α

text: is

text: the

text: i

text: unique

text: smallest

text: element

text: of

text: ω

text: The

text: method

text: is

text: described

text: in

text: Lemma

text: P

text: (B)

text: Turn

text: around:

text: We

text: change

text: coordinates

text: from

text: one

text: open

text: aﬃne

text: chart

text: of

text: to

text: an-

text: other

text: by

text: replacing

text: x

text: by

text: As

text: a

text: result,

text: val(α

text: val(α

text: and

text: the

text: relative

text: order

text: i

text: i

text: of

text: the

text: valuations

text: on

text: the

text: tuple

text: α

text: is

text: reversed

text: on

text: the

text: new

text: tuple

text: α

text: Lemma

text: [Separating

text: points]

text: Consider

text: a

text: repeated

text: coordinate

text: ω

text: of

text: ω,

text: and

text: write

text: β

text: max

text: val(α

text: α

text: ω

text: ω

text: ω

text: for

text: m

text: l

text: ω.

text: m

text: l

text: m

text: l

text: Fix

text: two

text: indices

text: i,

text: j

text: with

text: ω

text: =ω

text: =ω

text: and

text: β

text: val(α

text: α

text: If

text: in(α

text: α

text: ζ

text: K

text: for

text: some

text: i

text: j

text: i

text: j

text: i−

text: j

text: ζ

text: with

text: val(ζ)

text: choose

text: γ

text: val(K

text: with

text: β

text: γ

text: val(α

text: α

text: ζtβ).

text: Then,

text: the

text: linear

text: P

text: P

text: i

text: j

text: change

text: of

text: coordinates

text: ψ

text: deﬁned

text: locally

text: by

text: ψ(x)

text: x

text: α

text: ζtβ

text: tγ

text: j

text: turns

text: the

text: tuple

text: α

text: (K

text: into

text: α

text: (K

text: where

text: their

text: coordinatewise

text: negative

text: valua-

text: tions

text: ω

text: and

text: ω

text: satisfy

text: the

text: following

text: properties:

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: 14 MARIA ANGELICA CUETO AND HANNAH MARKWIG Figure 6. Visualizing the coordinate change (A) via tropical modiﬁcations. (cid:48) (1) ω = ω > ω if ω > ω ; s(cid:48) s i (cid:48)s −i (2) ω = ω , and in(α ) = in(α ) if ω < ω ; s(cid:48) −i (cid:48) −s − i ≤ s i (cid:54) (3) ω = γ < ω = val(α α ) ω if ω = ω and s = i. i s s i i s i Proof. The ﬁrst claim follows immediately from the strong non-Archimedean inequality − since α + ζtβ + tγ has valuation ω . A similar argument proves the second claim. In j (cid:48) (cid:54) (cid:54) i particular, α = 0 whenever ω = ω . s s i − − (cid:48) (cid:48) (cid:54) We now prove the third item. Again, val(α α ζtβ) > γ, so val(α ) = γ and α = 0. (cid:54) i j i i Pick s = i with ω = ω . We wr(cid:124)ite (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) s i (cid:48) − − − − α = (α α ) + (α α ζtβ) tγ . s s i i j − ≤ · ≤ · ωi val( ) β val( )=γ>β − ≤ (cid:48) − (cid:48) (cid:54) (cid:3) By the strong non-Archimedean inequality, ω val(α )=val(α α ) < γ, so α =0. i s s i s The eﬀect of the coordinate change (A) from Lemma 5.3 can easily be visualized by means of a tropical modiﬁcation followed by a coordinate projection, as we now illustrate: C{{ }} Example 5.4. Consider points in the Puiseux series ﬁeld K = t : − − − ∗ α = t3, α = 2 + t, α = 2 + t2, α = t 2, α = t 3, and α = t 4 in K , 1 2 3 4 5 6 − where ω = 3, ω = ω = ω = 0, ω = 2, ω = 3, ω = 4, β =ζ =1, 1 < γ = 3/2 < val(t2). 1 2 3 4 − 5 6 R To separate α from α , and place val(α ) to the very left of , we reembed the line 2 − 3 − − 2 in the plane via y = x (2 + t2) t t3/2. The tropicalization of this planar line together (cid:5) with its marked points and the projection to the y-coordinate is depicted in Figure 6. P Our ﬁrst combinatorial procedure uses a change of coordinates in 1 and a relabeling (cid:48) to produce a new tuple α from α with the additional property that the maximum and (cid:48) minimum values of ω are attained exactly once. This is the content of Algorithm 1. In turn, Algorithm 2 transforms the output of Algorithm 1 into a conﬁguration in a suitable region Ω(i). We measure improvement by two auxiliary variables: • the deﬁciency def(α) of the point conﬁguration deﬁned as the size of the partition { } of [6] = 1, . . . , 6 identifying equal coordinates of ω, • a reﬁned partition Λ taking both the valuation and the initial terms into account. { } { } Our partitions will always have the singletons 1 and 6 since ω and ω remain isolated 1 6 after each iteration of Step (A). Proof of Algorithm 2. If the input α is already in one of the desired regions Ω(i) for i in { } I, . . . , VII , the algorithm outputs the pair (α, i). If not, our deﬁnition of the partition Λ (cid:48) of α ensures that for each value of d < 6, after applying turn around transformation (B) − ≤ − followed by a relabeling of [6] (to satisfy val(α ) val(α ) for all j = 1, . . . , 6) we − − { j − j+1 } can assume that ω = ω > val(α α ) and val(α α ) : ω = ω = ω is maximized 2 3 2 3 i j i j 2 P at i = 2, j = 3. A case by case analysis shows that a change of coordinates (A) on 1 (cid:48) ∈ ∗ (cid:48) with ω =ω , i=2, and j =3 turns α to α (K )6 and def(α ) > def(α). After each such 2 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: MARIA

text: ANGELICA

text: CUETO

text: AND

text: HANNAH

text: MARKWIG

text: Figure

text: Visualizing

text: the

text: coordinate

text: change

text: (A)

text: via

text: tropical

text: modiﬁcations.

label: list

text: ω

text: ω

text: ω

text: if

text: ω

text: ω

text: s

text: i

text: −i

text: ω

text: ω

text: and

text: in(α

text: in(α

text: if

text: ω

text: ω

text: −i

text: −s

text: i

text: s

text: i

text: ω

text: γ

text: ω

text: val(α

text: α

text: ω

text: if

text: ω

text: ω

text: and

text: s

text: i.

text: i

text: s

text: s

text: i

text: i

text: s

text: i

text: Proof.

text: The

text: ﬁrst

text: claim

text: follows

text: immediately

text: from

text: the

text: strong

text: non-Archimedean

text: inequality

text: since

text: α

text: ζtβ

text: tγ

text: has

text: valuation

text: ω

text: A

text: similar

text: argument

text: proves

text: the

text: second

text: claim.

text: In

text: j

text: i

text: particular,

text: α

text: whenever

text: ω

text: ω

text: s

text: s

text: i

text: We

text: now

text: prove

text: the

text: third

text: item.

text: Again,

text: val(α

text: α

text: ζtβ)

text: γ,

text: so

text: val(α

text: γ

text: and

text: α

text: i

text: j

text: i

text: i

text: Pick

text: s

text: i

text: with

text: ω

text: ω

text: We

text: s

text: i

text: α

text: (α

text: α

text: (α

text: α

text: ζtβ)

text: tγ

text: s

text: s

text: i

text: i

text: j

text: ωi

text: val(

text: β

text: val(

text: )=γ>β

text: By

text: the

text: strong

text: non-Archimedean

text: inequality,

text: ω

text: val(α

text: )=val(α

text: α

text: γ,

text: so

text: α

text: i

text: s

text: s

text: i

text: s

text: The

text: eﬀect

text: of

text: the

text: coordinate

text: change

text: (A)

text: from

text: Lemma

text: can

text: easily

text: be

text: visualized

text: by

text: means

text: of

text: a

text: tropical

text: modiﬁcation

text: followed

text: by

text: a

text: coordinate

text: projection,

text: as

text: we

text: now

text: illustrate:

text: C{{

text: Example

text: Consider

text: points

text: in

text: the

text: Puiseux

text: series

text: ﬁeld

text: K

text: t

text: α

text: α

text: t,

text: α

text: α

text: t

text: α

text: t

text: and

text: α

text: t

text: in

text: K

text: where

text: ω

text: ω

text: ω

text: ω

text: ω

text: ω

text: ω

text: β

text: =ζ

text: γ

text: R

text: To

text: separate

text: α

text: from

text: α

text: and

text: place

text: val(α

text: to

text: the

text: very

text: left

text: of

text: we

text: reembed

text: the

text: line

text: in

text: the

text: plane

text: via

text: y

text: x

text: t

text: The

text: tropicalization

text: of

text: this

text: planar

text: line

text: together

text: with

text: its

text: marked

text: points

text: and

text: the

text: projection

text: to

text: the

text: y-coordinate

text: is

text: depicted

text: in

text: Figure

text: P

text: Our

text: ﬁrst

text: combinatorial

text: procedure

text: uses

text: a

text: change

text: of

text: coordinates

text: in

text: and

text: a

text: relabeling

text: to

text: produce

text: a

text: new

text: tuple

text: α

text: from

text: α

text: with

text: the

text: additional

text: property

text: that

text: the

text: maximum

text: and

text: minimum

text: values

text: of

text: ω

text: are

text: attained

text: exactly

text: once.

text: This

text: is

text: the

text: content

text: of

text: Algorithm

text: In

text: turn,

text: Algorithm

text: transforms

text: the

text: output

text: of

text: Algorithm

text: into

text: a

text: conﬁguration

text: in

text: a

text: suitable

text: region

text: Ω(i).

text: We

text: measure

text: improvement

text: by

text: two

text: auxiliary

text: variables:

text: the

text: deﬁciency

text: def(α)

text: of

text: the

text: point

text: conﬁguration

text: deﬁned

text: as

text: the

text: size

text: of

text: the

text: partition

text: of

text: identifying

text: equal

text: coordinates

text: of

text: ω,

text: a

text: reﬁned

text: partition

text: Λ

text: taking

text: both

text: the

text: valuation

text: and

text: the

text: initial

text: terms

text: into

text: account.

text: Our

text: partitions

text: will

text: always

text: have

text: the

text: singletons

text: and

text: since

text: ω

text: and

text: ω

text: remain

text: isolated

text: after

text: each

text: iteration

text: of

text: Step

text: (A).

text: Proof

text: of

text: Algorithm

text: If

text: the

text: input

text: α

text: is

text: already

text: in

text: one

text: of

text: the

text: desired

text: regions

text: Ω(i)

text: for

text: i

text: in

text: I,

text: VII

text: the

text: algorithm

text: outputs

text: the

text: pair

text: (α,

text: i).

text: If

text: not,

text: our

text: deﬁnition

text: of

text: the

text: partition

text: Λ

text: of

text: α

text: ensures

text: that

text: for

text: each

text: value

text: of

text: d

text: after

text: applying

text: turn

text: around

text: transformation

text: (B)

text: followed

text: by

text: a

text: relabeling

text: of

text: (to

text: satisfy

text: val(α

text: val(α

text: for

text: all

text: j

text: we

text: j

text: can

text: assume

text: that

text: ω

text: ω

text: val(α

text: α

text: and

text: val(α

text: α

text: ω

text: ω

text: ω

text: is

text: maximized

text: i

text: j

text: i

text: j

text: P

text: at

text: i

text: j

text: A

text: case

text: by

text: case

text: analysis

text: shows

text: that

text: a

text: change

text: of

text: coordinates

text: (A)

text: on

text: with

text: ω

text: =ω

text: and

text: j

text: turns

text: α

text: to

text: α

text: (K

text: and

text: def(α

text: def(α).

text: After

text: each

text: such

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text: TROPICAL GEOMETRY OF GENUS TWO CURVES 15 Algorithm 1: Separate the mininum and maximum values of ω. ∗ Input: A tuple α = (α , . . . , α ) of six distinct labeled points in K . 1 6 (cid:55)→ Assumption: The valuation on K splits via ω tω. (cid:48) P Output: A tuple α obtained from α by a linear change of coordinates in 1 followed − (cid:48) − (cid:48) ≤ ≤− (cid:48) − (cid:48) by a relabeling, where val(α )< val(α ) . . . val(α )< val(α ). − ≤ −1 ≤ 2 ≤ − ≤5− 6 Relabel the points so that val(α ) val(α ) . . . val(α ) val(α ); (cid:48) ← (cid:48) ← − (cid:48) − 1 (cid:48) − 2 (cid:48) 5 6 α α ; ω val(α ) := ( val(α ), . . . , val(α )); ← { (cid:48) } ←1{ 6 (cid:48) } Λ i: ω = min(ω ) ; Λ i: ω = max(ω ) ; mi|n | i max i if Λ > 1 then min { − ∈ (cid:54) } Relabel Λ so that val(α α ): i, j Λ , i = j is maximized at i=1, j =2; (cid:48) ← min i j min α ψ(α) where ψ is deﬁned as in (5.3) with ω = ω , i = 1, j = 2; (cid:48) ← − (cid:48) 1 ω val(α ); | | if Λ > 1 then m(cid:48) ←ax P (cid:48) ← α Coordinate change (B) on 1 applied to α ; Λ Λ ; { (cid:48) − (cid:48) ∈ (cid:54) }max min Relabel Λ so that val(α α ): i, j Λ , i = j is maximized for i=6, j =5; (cid:48) ← (cid:48)max i j max (cid:48) α ψ(α ) where ψ is deﬁned as in (5.3) with ω = ω , i = 6, j = 5; (cid:48) ← P (cid:48) 6 α Coordinate change (B) on 1 applied to α ; (cid:48) return α . − transformation, a relabeling of [6] is perform to ensure val(α ) is ordered increasingly, (cid:48) i (cid:48) and a new iteration is performed with the new input α and the new deﬁciency d = def(α ) (cid:3) while d < 6. The process stops in at most four steps. 6. Faithful re-embedding of planar hyperelliptic curves Up to this point, we have only dealt with abstract tropical curves. In this section, we turn our attention to embedded tropical plane curves, deﬁned as the dual complex to Newton subdivisions of (1.1) [11, 39, 45]. Our objective is to prove Theorem 1.2. Our ﬁrst result allows us to assume that the hyperelliptic cover (1.1) is branched at ∞ both 0, and , and take the leading coeﬃcient u to be 1. It ensures that the descriptions −∞ ∞ of witnesses regions from Table 1 remains valid in this setting, for ω = and ω = : P (cid:48) 1 6 Lemma 6.1. After an automorphism of 1 sending α to α, the equation (1.1) becomes (cid:89) 5 − − ∈ ∗ (6.1) g(x, y) := y2 x (x α ) = 0 α , . . . , α K , i 2 5 i=2 − ∈ R (cid:48) − (cid:48) (cid:48) − (cid:48) (cid:48) (cid:48) where ω := val(α ) , ω = ω 2ω = 2 val((cid:0)α ) val(α ), in α(cid:1) = in α / in α 2 for i i ≤ ≤ i ≤i 6 6 i i i 6 all i = 2, . . . , 5, and ω2 ω3 ω4 ω5. (cid:113)(cid:81) (cid:55)→ − (cid:48) (cid:48) − (cid:48) − (cid:48) Proof. We use the transformation ϕ: z (z α )/ (α α )(z α ) . Expression (6.1) 1 1 6 6 (cid:48) − − (cid:48) − (cid:48) (cid:48) − (cid:48) (cid:3) is obtained by replacing y with y = y(x 1/(α α ))3/ 5 (α α ). 1 6 k=1 k 6 As discussed in Section , the na¨ıve tropicalization Trop V (g) induced by (6.1) is almost G never faithful. Our goal in this section is to produce faithful re-embeddings in 3 for m all seven witness regions. We will make full use of the techniques developed in Section . Except for Type (II), faithfulness can be achieved in the XZ-plane, since the relative interior of the cell σ from (3.4) will contain no point from the re-embedded tropical curve 4 Trop V (I ). Furthermore, a reﬁned algebraic lift of the tropical polynomial F from (3.3) g,f ##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: TROPICAL

text: GEOMETRY

text: OF

text: GENUS

text: TWO

text: CURVES

text: Algorithm

text: Separate

text: the

text: mininum

text: and

text: maximum

text: values

text: of

text: ω.

text: Input:

text: A

text: tuple

text: α

text: (α

text: α

text: of

text: six

text: distinct

text: labeled

text: points

text: in

text: K

text: Assumption:

text: The

text: valuation

text: on

text: K

text: splits

text: via

text: ω

text: tω.

text: P

text: Output:

text: A

text: tuple

text: α

text: obtained

text: from

text: α

text: by

text: a

text: linear

text: change

text: of

text: coordinates

text: in

text: followed

text: by

text: a

text: relabeling,

text: where

text: val(α

text: val(α

text: val(α

text: val(α

text: −1

text: ≤5−

text: Relabel

text: the

text: points

text: so

text: that

text: val(α

text: val(α

text: val(α

text: val(α

text: α

text: α

text: ω

text: val(α

text: val(α

text: val(α

text: Λ

text: i:

text: ω

text: min(ω

text: Λ

text: i:

text: ω

text: max(ω

text: mi|n

text: i

text: max

text: i

text: if

text: Λ

text: then

text: min

text: Relabel

text: Λ

text: so

text: that

text: val(α

text: α

text: i,

text: j

text: Λ

text: i

text: j

text: is

text: maximized

text: at

text: j

text: min

text: i

text: j

text: min

text: α

text: ψ(α)

text: where

text: ψ

text: is

text: deﬁned

text: as

text: in

text: with

text: ω

text: ω

text: i

text: j

text: ω

text: val(α

text: if

text: Λ

text: then

text: ←ax

text: P

text: α

text: Coordinate

text: change

text: (B)

text: on

text: applied

text: to

text: α

text: Λ

text: Λ

text: }max

text: min

text: Relabel

text: Λ

text: so

text: that

text: val(α

text: α

text: i,

text: j

text: Λ

text: i

text: j

text: is

text: maximized

text: for

text: j

text: i

text: j

text: max

text: α

text: ψ(α

text: where

text: ψ

text: is

text: deﬁned

text: as

text: in

text: with

text: ω

text: ω

text: i

text: j

text: P

text: α

text: Coordinate

text: change

text: (B)

text: on

text: applied

text: to

text: α

text: return

text: α

text: transformation,

text: a

text: relabeling

text: of

text: is

text: perform

text: to

text: ensure

text: val(α

text: is

text: ordered

text: increasingly,

text: i

text: and

text: a

text: new

text: iteration

text: is

text: performed

text: with

text: the

text: new

text: input

text: α

text: and

text: the

text: new

text: deﬁciency

text: d

text: def(α

text: while

text: d

text: The

text: process

text: stops

text: in

text: at

text: most

text: four

text: steps.

text: Faithful

text: re-embedding

text: of

text: planar

text: hyperelliptic

text: curves

text: Up

text: to

text: this

text: point,

text: we

text: have

text: only

text: dealt

text: with

text: abstract

text: tropical

text: curves.

text: In

text: this

text: section,

text: we

text: turn

text: our

text: attention

text: to

text: embedded

text: tropical

text: plane

text: curves,

text: deﬁned

text: as

text: the

text: dual

text: complex

text: to

text: Newton

text: subdivisions

text: of

text: Our

text: objective

text: is

text: to

text: prove

text: Theorem

text: Our

text: ﬁrst

text: result

text: allows

text: us

text: to

text: assume

text: that

text: the

text: hyperelliptic

text: cover

text: is

text: branched

text: at

text: both

text: and

text: and

text: take

text: the

text: leading

text: coeﬃcient

text: u

text: to

text: be

text: It

text: ensures

text: that

text: the

text: descriptions

text: of

text: witnesses

text: regions

text: from

text: Table

text: remains

text: valid

text: in

text: this

text: setting,

text: for

text: ω

text: and

text: ω

text: P

text: Lemma

text: After

text: an

text: automorphism

text: of

text: sending

text: α

text: to

text: α,

text: the

text: equation

text: becomes

text: g(x,

text: y)

text: x

text: (x

text: α

text: α

text: α

text: K

text: i

text: R

text: where

text: ω

text: val(α

text: ω

text: ω

text: 2ω

text: val(α

text: in

text: in

text: α

text: in

text: α

text: for

text: i

text: i

text: i

text: ≤i

text: i

text: i

text: i

text: all

text: i

text: and

text: ω2

text: ω3

text: ω4

text: Proof.

text: We

text: use

text: the

text: transformation

text: ϕ:

text: z

text: (z

text: α

text: (α

text: α

text: )(z

text: α

text: Expression

text: is

text: obtained

text: by

text: replacing

text: y

text: with

text: y

text: y(x

text: α

text: (α

text: α

text: k

text: As

text: discussed

text: in

text: Section

text: the

text: na¨ıve

text: tropicalization

text: Trop

text: V

text: (g)

text: induced

text: by

text: is

text: almost

text: G

text: never

text: faithful.

text: Our

text: goal

text: in

text: this

text: section

text: is

text: to

text: produce

text: faithful

text: re-embeddings

text: in

text: for

text: m

text: all

text: seven

text: witness

text: regions.

text: We

text: will

text: make

text: full

text: use

text: of

text: the

text: techniques

text: developed

text: in

text: Section

text: Except

text: for

text: Type

text: (II),

text: faithfulness

text: can

text: be

text: achieved

text: in

text: the

text: XZ-plane,

text: since

text: the

text: relative

text: interior

text: of

text: the

text: cell

text: σ

text: from

text: will

text: contain

text: no

text: point

text: from

text: the

text: re-embedded

text: tropical

text: curve

text: Trop

text: V

text: (I

text: Furthermore,

text: a

text: reﬁned

text: algebraic

text: lift

text: of

text: the

text: tropical

text: polynomial

text: F

text: from

text: g,f

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##


text: TROPICAL GEOMETRY OF GENUS TWO CURVES 29 C − We start with the cell . The monomials aﬀected are y4 (if b2 = b2), and y2, yz2 4 34 2 and z3 (if 3b2 = b2). From Figure 12 we see that lowering any of these 4 monomials 34 2 will have no eﬀect on the generic subdivision since these points were already unmarked (the unmarking of y4 was indicated in pink). Therefore, there will be a single Newton C subdivision for , namely the generic one. 4 C Special subdivisions on the cell are determined by the behavior of y3 whenever ± 5 b2 = b b . This monomial is marked in Qgen, as seen in Figure 12. When the height 4 5 34 5 of this monomial is reduced, an edge between yz and y4 arises. Furthermore, with the exception of y3, all the heights of all points in the triangle T with vertices y, yz and y4 are known from Table 3. Linear algebra reveals that depending on the height of y3, there will be two possible subdivisions: either T is a polygon in the subdivision, or it gets divided along an edge between y3 and yz. Numerical examples on the Supplementary material conﬁrm that both cases do occur. C C The cell has the same deﬁning genericity conditions as , with the addition that y4 6 4 drops height whenever y3 does. Since y2 is marked, the lowering of the monomials z3, y2z and yz will not change the subdivision, so we can disregard this genericity condition, and − only require b2 = b2 for special behavior. 34 2 Furthermore, since y3 and y4 are both marked in Qgen as we see in Figure 12, for special 6 parameters, an edge joining y2 and y2z2 will appear, and give rise to a triangle T with vertices y2, y2z2 and y5. We claim that T can only be further subdivided by an edge between y2z2 and y3 leading to the two possibility for Qsp shown in the ﬁgure. The reason 6 for this lies in Lemma 3.3 and Proposition 6.10. Indeed, by linear algebra, we know that ∩ v = (2ω + ω /2, ω + ω + ω /2) so this vertex lies in σ σ . Therefore, all cells in a 12 4 5 2 4 5 4 5 (cid:48) (cid:48)(cid:48) subdivision of T will come from vertices in σ , namely the vertices v and v in Figure 9. 5 34 34 Unless these two agree, the edge between them in Trop V (g˜) is dual to an edge with slope − 2 in a subdivision of T . By convexity, there is only one option for such an edge. C The analysis of non-genericity for the cells with i = 0, 1, 2 is simpler that earlier cases 7,i ± since only the monomial y3 imposes restrictions on the parameters. Only if b2 = b b this 4 2 5B monomial will be lower than expected. If so, due to the marking of y4 in the polygon gen from Figure 13, an edge between y2z and y4 will appear for special parameters. Depending on the height of y3, we will have one extra edge joining y2z and y3. This yields the two B possible conﬁgurations sp in the ﬁgure. C Finally, we discuss the subdivisions for non-generic parameters coming from . The C 8 same six monomials from are responsible for special choices of parameters. Since these 6 six monomials were not vertices in the generic subdivision in Figure 11, lowering them will not alter the subdivision, except for unmarking y3 and y4 accordingly. We conclude the C (cid:3) generic and the special Newton subdivisions agree for . This concludes our proof. 8 Formulas for all vertices in Trop V (I ) can be given in terms of the vertices v , v , v f,g 1 2 4 from (6.4) (where ω = ω ) and the weight vector (ω , ω , d , ω ) from Table 1: 3 4 5 4 34 2 − (cid:48) − (cid:48) v12 := v1 +− (ω5 − ω2)/2 (1, 1, 0), v1(cid:48)(cid:48)2 := v1(cid:48)(cid:48)2− ε (cid:48)(1, 1, 0), v := v (ω ω )/2 (1, 2, 3), v := v 2ε (1, 1, 3), 34 4 5 2 12 2 − − (cid:48) (cid:48) v := v (ω d )(0, 1, 1), v := v + 2ε(1, 2, 0), 20 2 4 34 34 2 (6.15) (cid:48)(cid:48) − − − (cid:48)(cid:48) − v21 := v2 (ω4 ω5 2d34)/2 (0, 1, 1) v34 := v34 ε(1, 2, 3), (cid:48) − − (cid:48) (cid:48)(cid:48) − v2 := v2 (ω4 ω2)(0, 0, 1), v21 := v21 + ε (0, 1, 1), (cid:48)(cid:48) − − (cid:48)(cid:48) (cid:48) − (cid:48)(cid:48) v := v (ω ω )(0, 0, 1), v := v 2ε (0, 1, 2), 2 2 5 4 21 2

text: TROPICAL

text: GEOMETRY

text: OF

text: GENUS

text: TWO

text: CURVES

text: C

text: We

text: start

text: with

text: the

text: cell

text: The

text: monomials

text: aﬀected

text: are

text: (if

text: and

text: and

text: (if

text: From

text: Figure

text: we

text: see

text: that

text: lowering

text: any

text: of

text: these

text: monomials

text: will

text: have

text: no

text: eﬀect

text: on

text: the

text: generic

text: subdivision

text: since

text: these

text: points

text: were

text: already

text: unmarked

text: (the

text: unmarking

text: of

text: was

text: indicated

text: in

text: pink).

text: Therefore,

text: there

text: will

text: be

text: a

text: single

text: Newton

text: C

text: subdivision

text: for

text: namely

text: the

text: generic

text: one.

text: C

text: Special

text: subdivisions

text: on

text: the

text: cell

text: are

text: determined

text: by

text: the

text: behavior

text: of

text: whenever

text: b

text: b

text: This

text: monomial

text: is

text: marked

text: in

text: Qgen,

text: as

text: seen

text: in

text: Figure

text: When

text: the

text: height

text: of

text: this

text: monomial

text: is

text: reduced,

text: an

text: edge

text: between

text: yz

text: and

text: arises.

text: Furthermore,

text: with

text: the

text: exception

text: of

text: all

text: the

text: heights

text: of

text: all

text: points

text: in

text: the

text: triangle

text: T

text: with

text: vertices

text: y,

text: yz

text: and

text: are

text: known

text: from

text: Table

text: Linear

text: algebra

text: reveals

text: that

text: depending

text: on

text: the

text: height

text: of

text: there

text: will

text: be

text: two

text: possible

text: subdivisions:

text: either

text: T

text: is

text: a

text: polygon

text: in

text: the

text: subdivision,

text: or

text: it

text: gets

text: divided

text: along

text: an

text: edge

text: between

text: and

text: yz.

text: Numerical

text: examples

text: on

text: the

text: Supplementary

text: material

text: conﬁrm

text: that

text: both

text: cases

text: do

text: occur.

text: C

text: C

text: The

text: cell

text: has

text: the

text: same

text: deﬁning

text: genericity

text: conditions

text: as

text: with

text: the

text: addition

text: that

text: drops

text: height

text: whenever

text: does.

text: Since

text: is

text: marked,

text: the

text: lowering

text: of

text: the

text: monomials

text: and

text: yz

text: will

text: not

text: change

text: the

text: subdivision,

text: so

text: we

text: can

text: disregard

text: this

text: genericity

text: condition,

text: and

text: only

text: require

text: for

text: special

text: behavior.

text: Furthermore,

text: since

text: and

text: are

text: both

text: marked

text: in

text: Qgen

text: as

text: we

text: see

text: in

text: Figure

text: for

text: special

text: parameters,

text: an

text: edge

text: joining

text: and

text: will

text: appear,

text: and

text: give

text: rise

text: to

text: a

text: triangle

text: T

text: with

text: vertices

text: and

text: We

text: claim

text: that

text: T

text: can

text: only

text: be

text: further

text: subdivided

text: by

text: an

text: edge

text: between

text: and

text: leading

text: to

text: the

text: two

text: possibility

text: for

text: Qsp

text: shown

text: in

text: the

text: ﬁgure.

text: The

text: reason

text: for

text: this

text: lies

text: in

text: Lemma

text: and

text: Proposition

text: Indeed,

text: by

text: linear

text: algebra,

text: we

text: know

text: that

text: v

text: ω

text: ω

text: ω

text: ω

text: so

text: this

text: vertex

text: lies

text: in

text: σ

text: σ

text: Therefore,

text: all

text: cells

text: in

text: a

text: subdivision

text: of

text: T

text: will

text: come

text: from

text: vertices

text: in

text: σ

text: namely

text: the

text: vertices

text: v

text: and

text: v

text: in

text: Figure

text: Unless

text: these

text: two

text: agree,

text: the

text: edge

text: between

text: them

text: in

text: Trop

text: V

text: (g˜)

text: is

text: dual

text: to

text: an

text: edge

text: with

text: slope

text: in

text: a

text: subdivision

text: of

text: T

text: By

text: convexity,

text: there

text: is

text: only

text: one

text: option

text: for

text: such

text: an

text: edge.

text: C

text: The

text: analysis

text: of

text: non-genericity

text: for

text: the

text: cells

text: with

text: i

text: is

text: simpler

text: that

text: earlier

text: cases

text: since

text: only

text: the

text: monomial

text: imposes

text: restrictions

text: on

text: the

text: parameters.

text: Only

text: if

text: b

text: b

text: this

text: monomial

text: will

text: be

text: lower

text: than

text: expected.

text: If

text: so,

text: due

text: to

text: the

text: marking

text: of

text: in

text: the

text: polygon

text: gen

text: from

text: Figure

text: an

text: edge

text: between

text: and

text: will

text: appear

text: for

text: special

text: parameters.

text: Depending

text: on

text: the

text: height

text: of

text: we

text: will

text: have

text: one

text: extra

text: edge

text: joining

text: and

text: This

text: yields

text: the

text: two

text: B

text: possible

text: conﬁgurations

text: sp

text: in

text: the

text: ﬁgure.

text: C

text: Finally,

text: we

text: discuss

text: the

text: subdivisions

text: for

text: non-generic

text: parameters

text: coming

text: from

text: The

text: C

text: same

text: six

text: monomials

text: from

text: are

text: responsible

text: for

text: special

text: choices

text: of

text: parameters.

text: Since

text: these

text: six

text: monomials

text: were

text: not

text: vertices

text: in

text: the

text: generic

text: subdivision

text: in

text: Figure

text: lowering

text: them

text: will

text: not

text: alter

text: the

text: subdivision,

text: except

text: for

text: unmarking

text: and

text: accordingly.

text: We

text: conclude

text: the

text: C

text: generic

text: and

text: the

text: special

text: Newton

text: subdivisions

text: agree

text: for

text: This

text: concludes

text: our

text: proof.

text: Formulas

text: for

text: all

text: vertices

text: in

text: Trop

text: V

text: (I

text: can

text: be

text: given

text: in

text: terms

text: of

text: the

text: vertices

text: v

text: v

text: v

text: f,g

text: from

text: (where

text: ω

text: ω

text: and

text: the

text: weight

text: vector

text: (ω

text: ω

text: d

text: ω

text: from

text: Table

text: ε

text: v

text: v

text: (ω

text: ω

text: v

text: v

text: 2ε

text: v

text: v

text: (ω

text: d

text: v

text: v

text: ω5

text: ε

text: v

text: v

text: (ω

text: ω

text: v

text: v

text: 2ε


text: 30 MARIA ANGELICA CUETO AND HANNAH MARKWIG Figure 10. All possible subdivisions corresponding to weight vectors in C the cells for i = 0, . . . , 4. The polygon Q indicates the subidivision 0,i C 0,i of the polygon Q on . Unlabeled polygons correspond to false crossings. 0,i Blue vertices come from σ . The notation on the remaining vertices is 4 compatible with Table 2 and Figure 9. C C Figure 11. All possible subdivisions for the cells for i = 0, 1, 2 and . 2,i 8 Red (respectively pink) dots indicate marked (resp. unmarked) monomials whose behavior varies with the genericity conditions. C Figure 12. All possible subdivisions for the cells for i = 1, 3, 4, 5, 6. i The polygon Q gets subdivided diﬀerently on each cell. The subscript gen correspond to generic parameters β , β , β and β , whereas the superscript 5 4 34 2 sp indicate non-generic ones. As with Figure 9, dotted lines correspond to extra possible subdivisions. When absent, the corresponding vertices agree. C Figure 13. All possible subdivisions for the cells for i = 0, 1, 2. 7,i ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: MARIA

text: ANGELICA

text: CUETO

text: AND

text: HANNAH

text: MARKWIG

text: Figure

text: All

text: possible

text: subdivisions

text: corresponding

text: to

text: weight

text: vectors

text: in

text: C

text: the

text: cells

text: for

text: i

text: The

text: polygon

text: Q

text: indicates

text: the

text: subidivision

text: C

text: of

text: the

text: polygon

text: Q

text: on

text: Unlabeled

text: polygons

text: correspond

text: to

text: false

text: crossings.

text: Blue

text: vertices

text: come

text: from

text: σ

text: The

text: notation

text: on

text: the

text: remaining

text: vertices

text: is

text: compatible

text: with

text: Table

text: and

text: Figure

text: C

text: C

text: Figure

text: All

text: possible

text: subdivisions

text: for

text: the

text: cells

text: for

text: i

text: and

text: Red

text: (respectively

text: pink)

text: dots

text: indicate

text: marked

text: (resp.

text: unmarked)

text: monomials

text: whose

text: behavior

text: varies

text: with

text: the

text: genericity

text: conditions.

text: C

text: Figure

text: All

text: possible

text: subdivisions

text: for

text: the

text: cells

text: for

text: i

text: i

text: The

text: polygon

text: Q

text: gets

text: subdivided

text: diﬀerently

text: on

text: each

text: cell.

text: The

text: subscript

text: gen

text: correspond

text: to

text: generic

text: parameters

text: β

text: β

text: β

text: and

text: β

text: whereas

text: the

text: superscript

text: sp

text: indicate

text: non-generic

text: ones.

text: As

text: with

text: Figure

text: dotted

text: lines

text: correspond

text: to

text: extra

text: possible

text: subdivisions.

text: When

text: absent,

text: the

text: corresponding

text: vertices

text: agree.

text: C

text: Figure

text: All

text: possible

text: subdivisions

text: for

text: the

text: cells

text: for

text: i

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: 36 MARIA ANGELICA CUETO AND HANNAH MARKWIG Remark 7.4. The previous result shows that the three tropical Igusa functions do not characterize tropical curves of genus two, not even within a ﬁxed combinatorial type and should not be considered tropical analogs of Igusa invariants. Indeed, the second and third tropical Igual functions agree on each cone in Mtrop, and all three agree on the cone of 2 theta curves. In particular, we cannot recover the length data for each tropical curve from these three functions. Proof of Theorem 7.3. Consider a generic lift X of our tropical curve. We ﬁx the six branch ∗ points of X to be α , . . . , α in K . The tropicalization of X is uniquely determined by the 1 6 valuation of the branch points, and their diﬀerences. Likewise, the formulas for the lengths L , L , L in Table 1 are described in terms of these parameters. By expression (7.6), the 0 1 2  tropical Igusa functions of X equal:   − −  val(j1(X)) = 5 val(A) + val(D) , − − − (7.7) val(j (X)) = 3 val(A) val(B) + val(D) , 2 − − − val(j (X)) = 2 val(A) val(C) + val(D). 3 By Algorithms 1 and 2, we may assume that the negative valuations of the branch points satisfy one of the seven conditions described in Table 1. Furthermore, we assume val(α ) = 0, and u=1 since u plays no role when deﬁning each j (X). 1 i We treat each invariant separately, analyzing the contributions of each summand in the deﬁnition of the four polynomials A, B, C, and D, and checking for potential cancellations of the expected initial terms. The proof is completed by discussing the behavior of each maximal cells of Mtrop separately in two lemmas below. 2 The genericity conditions on X are imposed so that the initial forms of each polynomial have the expected valuation after specializing them at the initial forms of each branch point. The two maximal cells in Mtrop require no genericity assumptions, since the leading 2 terms of all polynomials involved are monomials. Type (I) cells: For a Type (I) curve, the negative valuation of each A, B, C, D is obtained by computing the initial term on each of these four polynomials with respect to the weight ∈ R vector ω := (ω , . . . , ω ) 6 with ω < . . . < ω . Lemma 7.5 ensures that 1 6 1 6 − − val(A) = 2(ω + ω + ω ), val(C) = 6(ω + ω ) + 4ω + 2ω , (7.8) − 4 5 6 − 5 6 4 3 val(B) = 4(ω +ω ) + 2(ω +ω ), val(D) = 2ω + 4ω + 6ω + 8ω + 10 ω . 5 6 3 4 2 3 4 5 6 Combining these values with (7.7) and the formulas for L , L and L from Table 1 0 1 2 gives: − jtrop(Trop X) = 10(ω + ω + ω ) (2ω + 4ω + 6ω + 8ω + 10ω ) 1 4 5 6 2 3 4 5 6 − − − − − = 4ω + 2ω 2ω 4ω = 2(ω ω ) + 6(ω ω ) + 2(ω ω ) 4 5 2 3 5 4 4 3 3 2 = L + 12L + L , 1 0 2 − jtrop(Trop X) = 6(ω +ω +ω ) + 4(ω +ω ) + 2(ω +ω ) (2ω +4ω +6ω +8ω +10ω ) 2 4 5 6 5 6 3 4 2 3 4 5 6 − − − − − = 2ω + 2ω 2ω 2ω = 2(ω ω ) + 4(ω ω ) + 2(ω ω ) 5 4 3 2 5 4 4 3 3 2 = L + 8L + L , 1 0 2 − jtrop(Trop X) = 4(ω +ω +ω ) + 6(ω +ω )+ 4 ω + 2 ω (2ω +4ω +6ω +8ω +10ω ) 3 4 5 6 6 5 4 3 2 3 4 5 6 = L + 8L + L = jtrop(Trop X) . 1 0 2 2 ##LTLine##

text: MARIA

text: ANGELICA

text: CUETO

text: AND

text: HANNAH

text: MARKWIG

text: Remark

text: The

text: previous

text: result

text: shows

text: that

text: the

text: three

text: tropical

text: Igusa

text: functions

text: do

text: not

text: characterize

text: tropical

text: curves

text: of

text: genus

text: two,

text: not

text: even

text: within

text: a

text: ﬁxed

text: combinatorial

text: type

text: and

text: should

text: not

text: be

text: considered

text: tropical

text: analogs

text: of

text: Igusa

text: invariants.

text: Indeed,

text: the

text: second

text: and

text: third

text: tropical

text: Igual

text: functions

text: agree

text: on

text: each

text: cone

text: in

text: Mtrop,

text: and

text: all

text: three

text: agree

text: on

text: the

text: cone

text: of

text: theta

text: curves.

text: In

text: particular,

text: we

text: cannot

text: recover

text: the

text: length

text: data

text: for

text: each

text: tropical

text: curve

text: from

text: these

text: three

text: functions.

text: Proof

text: of

text: Theorem

text: Consider

text: a

text: generic

text: lift

text: X

text: of

text: our

text: tropical

text: curve.

text: We

text: ﬁx

text: the

text: six

text: branch

text: points

text: of

text: X

text: to

text: be

text: α

text: α

text: in

text: K

text: The

text: tropicalization

text: of

text: X

text: is

text: uniquely

text: determined

text: by

text: the

text: valuation

text: of

text: the

text: branch

text: points,

text: and

text: their

text: diﬀerences.

text: Likewise,

text: the

text: formulas

text: for

text: the

text: lengths

text: L

text: L

text: L

text: in

text: Table

text: are

text: described

text: in

text: terms

text: of

text: these

text: parameters.

text: By

text: expression

text: the

text: tropical

text: Igusa

text: functions

text: of

text: X

text: equal:

text: val(A)

text: val(D)

text: val(j

text: (X))

text: val(A)

text: val(B)

text: val(D)

text: val(j

text: (X))

text: val(A)

text: val(C)

text: val(D).

text: By

text: Algorithms

text: and

text: we

text: may

text: assume

text: that

text: the

text: negative

text: valuations

text: of

text: the

text: branch

text: points

text: satisfy

text: one

text: of

text: the

text: seven

text: conditions

text: described

text: in

text: Table

text: Furthermore,

text: we

text: assume

text: val(α

text: and

text: since

text: u

text: plays

text: no

text: role

text: when

text: deﬁning

text: each

text: j

text: (X).

text: i

text: We

text: treat

text: each

text: invariant

text: separately,

text: analyzing

text: the

text: contributions

text: of

text: each

text: summand

text: in

text: the

text: deﬁnition

text: of

text: the

text: four

text: polynomials

text: A,

text: B,

text: C,

text: and

text: D,

text: and

text: checking

text: for

text: potential

text: cancellations

text: of

text: the

text: expected

text: initial

text: terms.

text: The

text: proof

text: is

text: completed

text: by

text: discussing

text: the

text: behavior

text: of

text: each

text: maximal

text: cells

text: of

text: Mtrop

text: separately

text: in

text: two

text: lemmas

text: below.

text: The

text: genericity

text: conditions

text: on

text: X

text: are

text: imposed

text: so

text: that

text: the

text: initial

text: forms

text: of

text: each

text: polynomial

text: have

text: the

text: expected

text: valuation

text: after

text: specializing

text: them

text: at

text: the

text: initial

text: forms

text: of

text: each

text: branch

text: point.

text: The

text: two

text: maximal

text: cells

text: in

text: Mtrop

text: require

text: no

text: genericity

text: assumptions,

text: since

text: the

text: leading

text: terms

text: of

text: all

text: polynomials

text: involved

text: are

text: monomials.

text: Type

text: (I)

text: cells:

text: For

text: a

text: Type

text: (I)

text: curve,

text: the

text: negative

text: valuation

text: of

text: each

text: A,

text: B,

text: C,

text: D

text: is

text: obtained

text: by

text: computing

text: the

text: initial

text: term

text: on

text: each

text: of

text: these

text: four

text: polynomials

text: with

text: respect

text: to

text: the

text: weight

text: R

text: vector

text: ω

text: (ω

text: ω

text: with

text: ω

text: ω

text: Lemma

text: ensures

text: that

text: val(A)

text: ω

text: ω

text: val(C)

text: ω

text: 4ω

text: 2ω

text: val(B)

text: +ω

text: +ω

text: val(D)

text: 2ω

text: 4ω

text: 6ω

text: 8ω

text: ω

text: Combining

text: these

text: values

text: with

text: and

text: the

text: formulas

text: for

text: L

text: L

text: and

text: L

text: from

text: Table

text: gives:

text: jtrop(Trop

text: X)

text: ω

text: ω

text: 4ω

text: 6ω

text: 8ω

text: 4ω

text: 2ω

text: 2ω

text: 4ω

text: ω

text: ω

text: ω

text: L

text: L

text: jtrop(Trop

text: X)

text: +ω

text: +ω

text: +ω

text: +ω

text: 2ω

text: 2ω

text: 2ω

text: 2ω

text: ω

text: ω

text: ω

text: L

text: L

text: jtrop(Trop

text: X)

text: +ω

text: +ω

text: +ω

text: ω

text: ω

text: L

text: L

text: jtrop(Trop

text: X)

text: ##LTLine##