Currently I'm studying the article *[Moduli of Enriques surfaces and Grothendieck-Riemann-Roch](http://arxiv.org/abs/math/0701546)* 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.



Family of Enriques surfaces and Grothendieck-Riemann-Roch
This is a somewhat technical remark, related to Andrea's answer, which is a bit too big to fit into the comment box.


If $f: Y \rightarrow T$ has connected fibres, to conclude that
$R^0f\_\*\mathcal O\_Y = \mathcal O\_T$, one needs some assumptions beyond just that $f$ is a projective morphism of Noetherian schemes. (Consider these examples: a closed embedding will have connected fibres. To give such an example in which all fibres non-empty, consider a non-reduced $T$, and let $Y$ be the underlying reduced subscheme. Or one could take $T$ to be a cuspidal cubic curve and $Y$ to be its normalization.)


What the theorem on formal functions shows (assuming that $f$ is projective, and that $Y$ and $T$ are Noetherian, so we can apply the result as it is proved in Hartshorne) is that
for any point $P$ in $T$, the $\mathfrak m\_P$-adic completion $(R^0f\_\*\mathcal O\_Y\hat{)}\_P$
is equal to $H^0(\hat{Y}\_P,\mathcal O)$, the global sections of the structure sheaf on the
formal fibre $\hat{Y}\_P$ over $P$.


So if $f$ has connected fibres, and hence connected formal fibres, so that
 $H^0(\hat{Y}\\_P,\mathcal O)$ is a local ring, we see that $(R^0f\_\*\mathcal O\_Y\hat{)}\_P$ is a finite local $\hat{\mathcal O}\\_{T,P}$-algebra. In general, one can't do better than this.


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 $\mathfrak m\_P$ of $R^0f\_\*\mathcal O\_Y$ 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 $k(P)$.


So, maintaining these assumptions on $f$, we see that for each point $P$ of $T$,
the stalk $(R^0f\_\*\mathcal O\_Y)\_P$ 
is a finite $\mathcal O\\_{T,P}$-algebra with the property that its reduction modulo
$\mathfrak m\_P$ is isomorphic to
the residue field $k(P)$ of ${\mathcal O}\\_{T,P}$. This implies (by Nakayama)
 that the natural map ${\mathcal O}\\_P \rightarrow (R^0f\_\*\mathcal O\_Y)\\_P$
is surjective. This is true at every $P$, and so we see that
$\mathcal O\_T \rightarrow R^0f\_\*\mathcal O\_Y$ is surjective.


Now one can combine this with the Grauert result to conclude (since a surjection
of invertible sheaves is necessarily an isomorphism) that the natural map
 $\mathcal O\_T \rightarrow R^0f\_\*\mathcal O\_Y$ is an isomorphism. (We probably don't
need to use the full force of Grauert here; for example, suppose that $T$ is connected;
a flat map is open, and a projective map is closed, so $f$ is surjective, hence faithfully
flat, and this implies that the map $\mathcal O\_T \rightarrow R^0f\_\*\mathcal O\_Y$ is
injective, I think.) 


Added: See Keerthi Madapusi's answer below for a correction to the above discussion
of flat base-change.


