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For a smooth $Y$, a necessary condition for contractibility is that the conormal line bundle $N_{Y,X}^*$ is ample. It is also sufficient for contracting to an algebraic space. The reference is Algebra …

Q1: A very simple example is given in Grothendieck's Tohoku paper "Sur quelques points d'algebre homologique", sec. 3.8. Edit: The space is the plane, and the sheaf is constructed by using a union of …

1. Are there simple examples (say on a curve or surface) of line bundles that are globally generated but not ample, of ample line bundles with no sections, of ample line bundles that are globally gene …

From the exact sequence $$1\to O^*\to K^*\to K^* / O^*\to 1$$ you see that, for as long as $H^1(K^*)=0$, the map from $H^0( K^*/ O^*)$ (i.e. Cartier divisors) to $H^1( O^*)$ (i.e. line bundles) is …

About Jon Woolf's answer, it seems to me that the condition that "$x$ is a closed point" was implicitly used: the extension by zero $Z_A$ is only defined for a locally closed subset $A$ (see e.g. Tenn …

Searching for various examples and counterexamples for sheaves, it is sometimes helpful to look at partially ordered sets with the poset topology: a set $U$ is open if and only if $x \in U$ and $x < y …

Let $Df$ denote the derivative of a function $f(x)$ and $\bigtriangledown f=f(x)-f(x-1)$ be the discrete derivative. Using the Taylor series expansion for $f(x-1)$, we easily get $\bigtriangledown = …

This is an example when proving "locally free" instead of merely "flat" is easier and more straightforward, and no Noetherian assumption on the base is needed. The point is that if some coefficient $a …

A related question (but not exactly the one you asked) is: Can one tell if a convex polytope $P$ and its translations by $\mathbb Z^n$ tile $\mathbb R^n$? Which polytopes $P$ have this property? …

Yes, you (and BCnrd) are absolutely correct and the quoted statement is wrong. Over any scheme $S$, the $S$-points of $\mathbb P^n$ are the surjections $\mathcal O_S^{\oplus n+1} \to F$ with invertib …

This is to fill some prerequisites to BCnrd's comment-answer. First of all, there are several definitions of a polarization on an abelian variety, and the most "coordinate-free" one is that it is a h …

Small contractions are mirrors to degenerations, so: degenerate, then deform out.

Just to remark that for a rational polytope whose vertices are not integral, the function $f_P(t)$ could still be a polynomial (and not just a quasipolynomial). A large class of examples is provided b …

The dimensions $h^i(\mathcal O_X)$ of the cohomology groups of $\mathcal O_X$, and thus the Euler characteristic, are birational invariants of smooth proper varieties in positive characteristic as wel …

Merely observe that a toric variety is the union of torus orbits $(\mathbb C^\*)^r$ for various dimensions $r$, and that the Euler characteristic of $(\mathbb C^\*)^r$ is zero if $r>0$ and $1$ if $r=0 …