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You need to assume that $X$ is irreducible, otherwise, as Tom points out, there are easy counterexamples. On the other hand, when $X$ is irreducible the answer is positive. The following proof is som …

Suppose that $H$ is an hyperplane such that $Z = H \cap X$. Let $L$ be a subspace of codimension 1 of $H$; the pencil of hyperplanes passing through $L$ defines a morphism $X \smallsetminus L \to \mat …

It was proved by Gabber and De Jong that the cohomological Brauer group of a quasi-compact scheme with an ample invertible sheaf equals its Brauer group. Our preprint does not contain this result.

A) You can easily reduce to the case that $S$ is the spectrum of a DVR $R$. Furthermore, by passing to a finite extension of $R$, you can assume that the components of the closed fiber are geometrical …

William Lang produced examples of surfaces of general type in positive characteristic with non-zero vector fields. Since surfaces of general type must have finite automorphism groups, this gives examp …

This is correct in characteristic $0$. Let $G$ be the kernel of the action; then one has to show that $X_{R_L}/G$ is a stable curve over $R_L^G$. This is Lemma 3.2 in my paper with Dan Abramovich Comp …

Here is a counterexample with $V$ finite. Suppose that $k'$ is a separable extension of degree $5$ of $k$, and set $V = \mathop{\rm Spec} k \sqcup \mathop{\rm Spec} k'$. Suppose that $V$ is a homogen …

1 - This is true for locally factorial schemes; in this case all Weil divisors are Cartier, and extending Weil divisors is not a problem. 2 - This is not true in general (for example, if you glue tog …

When doing moduli theory over $\mathbb Z$, or another base scheme, one works with sheaves that are flat over the base; this implies that all the discrete invariants, such as the Hilbert polynomial, ar …

It does not have suprema. For example, consider the affine plane $\mathbb A^2_{\mathbb C}$, and let $\mathbb A^1_{\mathbb C}$ by the line $y=0$. Consider the locally closed subschemes $U := \mathbb A^ …

Take $g+1$ general points $p_1, \dots, p_{g+1}$ on your curve. The divisor $p_1+ \cdots +p_g - p_{g+1}$ is non-special. The proof is easy from the following lemma: if $D$ is a divisor such that $\math …

It it true for any $Y$: see Corollaire 21.12.7 of EGAIV.

Brian Conrad has written up a proof of Nagata's theorem, starting from notes of Deligne: http://math.stanford.edu/~conrad/papers/nagatafinal.pdf. About the analytic case, I have no idea.

How about smooth proper morphisms $X \to S$ with connected fibers, a section $S \to X$, such that the sheaf of Kähler differentials $\Omega_{X/S}$ is a pullback from $S$, and such that the group schem …

Yes, even when $X$, $C$ and $Y$ are as nasty as they can be. Chern classes are functorial.