14.295 ag.algebraic-geometry questions.

6 Sato-Tate and the angles of split primes

I was reading this blog by Evan Chen about complex multiplication. He's discussing Sato-Tate Conjecture. We can have elliptic curve $E/\mathbb{Q}$ and solve it over finite fields. \begin{eqnarray*} ...

12 Defining abstract varieties and their morphisms over a finitely generated subfield of the base field

Let $k$ be an algebraically closed field. By a finitely generated subfield of $k$ I mean a subfield $k_0\subset k$ that is finitely generated over the prime subfield of $k$ (that is, over $\mathbb Q$ ...

6 Cokernel of map of étale sheaves

Let $p:\mathbb{G}_m\to \operatorname{Spec} k$ be the structure map, and let $T$ be an algebraic $k$-torus viewed as an étale sheaf over $k$. Why is the cokernel of the canonical map $T\to p_*p^*T$ ...

1 Segre embedding and Hilbert polynomial of coherent sheaves

Let $X \subset \mathbb{P}^n$ and $Y \subset \mathbb{P}^m$ be smooth, projective subvarieties, $F$ and $G$ coherent, torsion-free, sheaves on $X$ and $Y$ with Hilbert polynomials $P_{F}$ and $P_G$, ...

2 References for Sheaves of Boolean Rings

I have been looking for references on sheaves that take value in the category of Boolean rings (e.g. about cohomology, etc). Would someone be able to give me some? Or are they interesting at all? ...

3 The ample cone of a surface with an algebraic $\mathbb C^*$ action

Let $X$ be a compact complex protective surface that admits a nontirvial algebraic $\mathbb C^*$-action. It seems to me, that the ample cone of $X$ is polyhedral with finite number of faces. I wonder ...

How to construct $\mathcal{H} ( \mathbb{P}^n ) = \{ \ \text{Smooth projective subvarieties of } \mathbb{P}^n \ \}$?

Set : $$\mathcal{H} ( \mathbb{P}^n ) = \{ \ \text{Smooth projective subvarieties of } \mathbb{P}^n \ \}$$ I would like to know if there exists a projective variety $H ( \mathbb{P}^n )$ whose ...

6 Cohomology of neighborhood of $\mathbb{C}\mathbb{P}^1$ in $\mathbb{C}\mathbb{P}^n$

Let $\mathbb{C}\mathbb{P}^1$ be embedded linearly to $\mathbb{C}\mathbb{P}^n$ with $n>1$. (Such an embedding is given in coordinates by $[x:y]\mapsto [x:y:0:\dots: 0]$.) Is it true that for any ...

2 Torsion free-ness of cohomology of moduli of vector bundles

My question requires a little introduction: $\textbf{Atiyah-Bott's solution:}$ In the paper "The Yang mills equation of Riemann surfaces" Atiyah-Bott has computed the cohomology of moduli vector ...

12 Brauer group of a curve over non-algebraically closed field

3 answers, 319 views ag.algebraic-geometry brauer-groups
It is a famous consequence of Tsen's theorem that a smooth curve over an algebraically closed field has trivial Brauer group. But what about curves over non algebraically closed fields? Let us fix a ...