9
votes

### Bott & Tu differential forms Example 10.1

In my version of Bott and Tu, the example reads (emphasis mine):
"Moreover, if $V \subset U$ is an inclusion of CONTRACTIBLE open sets, then $\rho^U_V: H^q(\pi^{-1} U) \to H^q( \pi^{-1} V)$ is an ...

8
votes

Accepted

### Bad prime of torsor and original elliptic curve ; Definition of Tate–Shafarevich group $Ш(E/K)$

I fear you wish for too much here.
If $Ш$ is finite, then we can represent each element by a torsor; each torsor has good reduction away from a finite set and the union of all bad places would then be ...

8
votes

Accepted

### Exceptional locus of rational map has codimension two

This not true. The simplest counterexample is the linear projection $Q \dashrightarrow \mathbb{P}^2$ from a smooth quadric surface $Q \subset \mathbb{P}^3$ from a point. The inverse map blows up two ...

6
votes

### What is the length of an algebraic curve?

Let me just clarify the proof Boris Bukh was thinking of in the question you link. Let $d$ be the degree of the curve, then the length of $Z \cap D_r(0)$ is at most $\pi r d$.
Consider a series of ...

5
votes

Accepted

### Is the derived support $\{x\in X\:|\: \mathsf{L}x^* M\neq 0\}$ closed?

No, in fact any subset of $X_0$ can be the support of a quasi-coherent sheaf on $X$. Indeed, suppose $A \subseteq X_0$ and consider the direct sum of skyscraper sheaves
$$M= \bigoplus_{x\in A} \kappa(...

5
votes

Accepted

### Normalisation of models of elliptic curves in finite extensions and reducedness of fibres

All isogenies have $\operatorname{ker}(f)$ defined over $K$, since it is the inverse image of a $K$-point (the identity) under a map of schemes defined over $K$ ($f$). However, not all isogenies are ...

4
votes

Accepted

### Terminal singularities of fibers vs total space

In your situation, $X$ has at most terminal singularities, at least when $X$ and $Y$ are complex varieties. To make the induction a bit easier, I will state the result as follows:
Theorem. Let $f \...

4
votes

### Relation between 16 $\mathbf{CP}^2$ and $\overline{K3}$

[Note: This answer was updated according to @MarcoGolla's comment to avoid a potential circularity.]
(1) The existence of such a bordism is already answered by your question: "$\mathbb{CP}^2$ ...

3
votes

Accepted

### Is the associated G/B fibration to a G-torsor projective?

A similar, but shorter answer. Choose a $G$-equivariant projective embedding $G/B \to \mathbb{P}(V)$. It gives a closed embedding
$$
G/B \times_G P \hookrightarrow \mathbb{P}(V) \times_G P
$$
over $X$...

2
votes

Accepted

### Making a map in sheaf cohomology involving a theta characteristic explicit

I claim that I can give a reasonably "geometric" formula.
Consider the surface $C \times C$ and the diagonal $\Delta$. The the line bundle $\vartheta \boxtimes \vartheta$ has vanishing ...

2
votes

### Flat base change and the quasicoherator

First, we note that the coherator on an affine scheme is given by the associated quasi-coherent sheaf to the global section (cf. Thomason-Trobaugh, Appendix B. 14).
Let $A$ be a ring.
Write $S :\...

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