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I am looking for examples of invertible sheaves in smooth, projective families such that the associated base locus (i.e., the intersection of all the effective divisors in the complete linear system) …

Let $\pi:X \to S$ be a flat, projective morphism, $S$ irreducible. Suppose that for all $s \in S$, the fiber $X_s$ satisfies $h^2(\mathcal{O}_{X_s})=0$. This means in particular that given an invertib …

Let $f:X \to Y$ be a projective morphism between noetherian scheme. Is there any know condition under which there exists a functorial stratification of $Y$ i.e., there exists a filtration by locally c …

The underlying field is $\mathbb{C}$. Let $\pi:\mathcal{C} \to \mathbb{A}^n$ be a flat family of projective curves (not necessarily smooth) of genus $g \ge 2$. Assume $\mathcal{C}$ is regular. Let $\m …

Let $k$ be an algebraically closed field of characteristic zero and $R$ a discrete valuation ring over $k$. Let $\pi:X \to \mathrm{Spec}(R)$ be a flat, projective morphism such that the generic fiber …

Let $f:X \to Y$ be a flat, projective morphism between projective varieties. Let $F, G$ be coherent sheaves on $X$, flat over $Y$. Let $\phi_1, \phi_2$ be two morphisms from $F$ to $G$ such that: 1) …

Are there examples of projective varieties over a non-algebraically closed field such that every geometrically stable sheaf on the variety is simple? I see, for example in Huybrechts-Lehn and in some …

Let $k$ be an infinite non-algebraically closed field, $X$ a smooth projective curve on $k$ and $E$ a locally-free sheaf on $X$ of rank at least $2$. Denote by $\bar{k}$ the algebraic closure of $k$, …

In the article "The irreducibility of the space of curves of given genus" by Deligne, Mumford, the definition of stable curves start with the assumption that the genus is at least $2$. Why is this nec …

Can someone suggest a reference to study Hodge locus, period mappings and period domains on moduli space of principally polarised abelian varieties? More precisely, consider the moduli space of princ …

Let $f:X \to Y$ be a birational morphism between projective, irreducible surfaces. Assume $X$ is non-singular and $Y$ is a hypersurface in $\mathbb{P}^3$ (not necessarily smooth). Is the arithmetic ge …

Let $X$ be a smooth projective variety (over the complex numbers) of dimension at least $2$, $B$ a finite set of closed points. Consider the closed subscheme $E:=B \times X + \Delta \subset X \times X …

Let $f:X \to Y$ be a surjective, smooth projective morphism of noetherian schemes. Let $\mathcal{L}$ be an inverible sheaf on $X$ satisfying $f_*\mathcal{L}$ is locally free and $s \in H^0(\mathcal{L} …

Let $R$ be a discrete valuation ring with residue field $k$, an algebraically closed field of characteristic zero and $\pi:X\to \mbox{spec}(R)$ a smooth, projective family of surfaces. Denote by $X_0 …

Let $R$ be a dvr, $X$ a flat, projective, integral, normal $R$-scheme such every closed fiber is again integral, normal. Let $F$ be a torsion-free coherent sheaf on $X$, flat over $R$. Is it true that …