TITLE: Family of Enriques surfaces and Grothendieck-Riemann-Roch
QUESTION [7 upvotes]: Currently I'm studying the article Moduli of Enriques surfaces and Grothendieck-Riemann-Roch by Pappas.
I am particularly interested in how he applies the GRR.
Q1. What is meant by a "family of Enriques surfaces"? I was guessing a flat morphism $f:Y\longrightarrow T$ of smooth projective varieties such that each fibre is an Enriques surface, but maybe this is too general?
Q2. In the article it says that the higher direct images $R^i f_\ast O_Y$ are zero ($i>0$). Is this an application of Grauert's theorem (III.12, Cor. 12.9, Hartshorne)? 
Q3. How to see easily that $R^0f_\ast O_Y = O_T$ ? I am guessing Grauert's theorem again.

REPLY [5 votes]: I couldn't figure out how to comment on Emerton's post, so I apologize for this non sequitur nitpick of an answer. 
"But, if $f$ is flat with geometrially connected and reduced fibres (e.g $f$ is smooth with geometrically connected fibres), then base-change for flat maps (Hartshorne III.9.3) shows that the fibre mod $P$ of $R^0fY$ is equal to $H^0(Y_P,\mathcal{O}_P)$ (the actual fibre over $P$, now, not the formal fibre), which equals $k(P)$ (the residue field at $P$), since $Y_P$ is projective, geometrically reduced, and geometrically connected over $k(P)$."
I am not sure that result from Hartshorne quite does that. It needs the map along which the base change is being made to be flat (since we need the Cech complex to remain exact under base change). In this case, we are pulling back along the decidedly unflat inclusion of the point $P$ into $Y$, and so would need the further technology of semi-continuity and base change from Hartshorne III.12 (which is unfortunately a piece of bad exposition, IMO).