ag.algebraic geometry - 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 $(R^0f_*mathcal O_Yhat{)}_P$
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 $(R^0f_*mathcal O_Yhat{)}_P$ 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.