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Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

2 votes
0 answers
35 views

Leray spectral sequence for étale homology

Let $X$, $Y$, $Z$ be quasi-projective varieties over an algebraically closed field $k$, $f: X \to Y$ and $g: Z \to X$ proper (even projective) maps with $f$ smooth, and $h: Z \to Y$ their composite. C …
3 votes
0 answers
62 views

Are motives of K3 surfaces of abelian type?

I refer to the article of van Geemen https://arxiv.org/pdf/math/9903146. What van Geemen calls the Kuga-Satake-Hodge conjecture suggests that for a K3 surface $X$ over $\mathbb{C}$, the summand $h^2(X …
3 votes
0 answers
140 views

Tate conjecture for singular varieties in terms of intersection homology

In his book “Mixed motives and algebraic K-theory”, Jannsen generalizes the Tate conjecture to a potentially singular projective variety $X$ over a finitely generated field. The statement is the same …
4 votes
1 answer
234 views

Known cases of Tate conjecture for varieties which are smooth over a curve

What are some examples of smooth projective varieties $X$ over a finite field for which the Tate conjecture for divisors is known, and which admit a smooth morphism to a smooth projective curve? I am …
1 vote
1 answer
158 views

Zeta function of variety over positive characteristic function field vs. local zeta factor o...

Let $X = Y \times_{\mathbb{F}_q} C$, with $Y, C / \mathbb{F}_q$ smooth projective varieties, $C$ a curve. Let $d = \dim_{\mathbb{F}_q} X$. We can consider the local zeta function $Z(X, t) = \prod\limi …
3 votes
Accepted

Zeta function of variety over positive characteristic function field vs. local zeta factor o...

The two zeta functions are the same. This is an immediate corollary of Milne, Etale Cohomology, proposition 13.8(c). Reference: Milne, J. S. Etale Cohomology (PMS-33). Princeton University Press, 1980 …
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  • 658
0 votes
0 answers
79 views

Potential typo in "Complete Systems of Two Addition Laws for Elliptic Curves" by Bosma and L...

Here is a link to the article: https://www.sciencedirect.com/science/article/pii/S0022314X85710888?ref=cra_js_challenge&fr=RR-1. Pages 237-238 give polynomial expressions $X_3^{(2)}, Y_3^{(2)}, Z_3^{( …
3 votes
0 answers
80 views

Let $f: X \to D$ be a proper holomorphic submersion. Does a holomorphic form on $X$, closed ...

Let $D$ be the unit disc in $\mathbb{C}^n$ and let $f: X \to D$ be a proper surjective holomorphic submersion, which is trivial as a smooth fiber bundle, with connected fiber $F$. We get an induced fl …
7 votes
1 answer
261 views

Explicit equations for the universal vector extension of an elliptic curve

The universal vector extension $E$ of an abelian variety $A$ is an algebraic group, an extension of $A$ by a vector group $0 \to V \to E \to A \to 0$, such that any other extension of $A$ by a vector …
3 votes
0 answers
178 views

Wondering if Monsky-Washnitzer ever published a result claimed to be forthcoming in a later ...

At the very end of the paper Formal Cohomology I by Monsky and Washnitzer, they write the following: "In some sense, the operator $\psi$ applied to a power series gives it "better growth conditions". …
1 vote

Does going-down theorem hold for local homomorphism of finite flat dimension?

Here's a counterexample with $A$ and $B$ of the same Krull dimension. Take $A = B = \mathbb{C}[x,y]_{(x,y)}$, and let $f: A \to B$ be the composition of the quotient of $A$ by $(y)$ with the embedding …
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4 votes
2 answers
316 views

Is the set of points on an abelian surface which project to rational points on the Kummer su...

Let $C$ be a hyperelliptic curve of genus 2 defined over $\mathbb{Q}$, let $J$ be its Jacobian, and let $X$ be the Kummer surface associated to $J$ (i. e. $X$ is the singular Kummer surface which resu …
2 votes
1 answer
218 views

Looking for an example of a point $P$ on an abelian variety $X$ such that no curve on $X$ co...

Is there an example of an abelian variety $X$ defined over a number field $K$, with $\dim X > 1$, and a $K$-rational point $P$ on $X$, such that no curve $C$ on $X$ (say defined over a number field) c …
1 vote

Any irreducible projective curve in $\mathbb P^3$ can be defined by three functions

EDIT: this answer has some issues. See the comments. I think a similar line of reasoning shows you can do it with four equations, though. For reasons of dimension your curve is contained in a hypersur …
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  • 658
3 votes
Accepted

Deligne finitude and finiteness of etale cohomology

See the answers here: https://mathoverflow.net/questions/76069/finiteness-of-étale-cohomology-groups . Because $k$ is finite, a constructible sheaf on (Spec $k)_{et}$ has finite cohomology groups. Del …
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  • 658

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