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Although Felipe answered your question, let me come back to David's wish, and give an answer using and advocating crystalline cohomology: First, if $X$ is, say, smooth and proper over a field $k$, t …

It also holds for minimal surfaces of Kodaira dimension $\kappa\geq0$.

Just to round out the picture: if the characteristic of $k$ is positive and $G$ is a finite, but non-reduced group scheme (for example, the infinitesimal group scheme $\mu_p$ of $p$.the roots of unity …

As already pointed out, the Hodge numbers may go up under reduction modulo $p$. On the other hand, let me also point out that the situation can be controlled: 1.) For all $p$, where $\overline{X}_p$ …

In case $B$ is ample this is Corollary 2.7 of Nori's paper: "Zariski's conjecture and related problems". I am not sure whether this result is originally due to Nori, but in this paper you will surely …

Quite generally, whenever you ``need a moduli space'', say, polarized deformations of varieties, or spaces of morphisms, you oftentimes construct it as follows: first, you construct some family in pro …

In order to have a contraction morphism $X\to Y$, the intersection matrix must be negative definite. Conversely, if the intersection matrix is negative definite, the contraction morphism exists in th …

Yes! However, as Rita, Torsten and Ru pointed out, my first ideas were too simple-minded. Although this makes the remarks by them somewhat unreadable, let me give the corrected answer: So, let $X$ be …

The adjunction formula in the form $$ \omega_C \cong \omega_X(C)|_C $$ holds whenever $C$ is a Cartier divisor on a Gorenstein scheme $X$. Taking Euler characteristics, you get an extremely general g …

degree of the canonical divisor doesn't make any sense as already pointed out by Mohammed. On the other hand, by "purity of the branch locus", the branch locus, as well as the ramification locus of …

I don't have an answer, but a couple of comments that may be useful: The group $H^1({\cal O}_X)$ can be identified with the Zariski tangent space of ${\rm Pic}^0(X)$. Since the latter can be non-redu …

Did you have a look at Satriano's article de Rham Theory for Tame Stacks and Schemes with Linearly Reductive Singularities?

By the way, for Enriques this more than just an example - it reflects their classification: the moduli space of Enriques surfaces is connected in any characteristic. All such surfaces arise as desingu …