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Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
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 …
4
votes
1
answer
183
views
Jacobian criterion for Zariski cotangent space over arbitrary field (X-post from SE)
I apologize as I am certain this is not research-level, but several days have gone by without an answer on stackexchange (https://math.stackexchange.com/questions/4724245/jacobian-criterion-for-zarisk …
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 …
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 …
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 …
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". …
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 …
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 …
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 …
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 …
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 …
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 …
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 …
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 …