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Questions tagged [ag.algebraic-geometry]

for questions on algebraic geometry, including algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

2
votes
1answer
174 views

How to produce low genus curves on abelian surfaces?

I would like to find "simple" complex algebraic curves (i.e. low genus) on a complex abelian surface $A$ (which are not just abelian subvarieties or translates of them). For example, a genus 2 curve, ...
8
votes
0answers
234 views

Status of the Sarkisov program a la Corti

To a birational map $f$ between Mori fiber spaces, one can associate the "Sarkisov degree", a triple $(\mu,\lambda,e)$ which I will not define here. To factor $f$ into a sequence of elementary ...
17
votes
1answer
609 views

Embedding abelian varieties into projective spaces of small dimension

Given a (complex) abelian variety $A$ of a fixed dimension $g$, let $d(A)$ be the dimension of the smallest complex projective space it embeds into. Is $d(A)$ uniform over all abelian varieties of a ...
4
votes
0answers
116 views

Nomality of blow-ups of higher dimensional rational singularities

A theorem (Theorem 8.1) of Lipman Lipman1969 shows that the blow-up of a rational singularity on a normal surface is still normal. Does anyone know under what (extra) condition Lipman's results can be ...
3
votes
0answers
127 views

A finitely generated ring as the limit of a category of finite rings?

I was reading the book of McLane about Categories. In Yoneda's lemma he shows that a functor $F$ evaluated in a object $O$ can be written as the set of natural transformations between the ...
7
votes
1answer
273 views

Is the quotient presheaf $\mathbb{G}_m/\mu_p$ an étale sheaf?

I apologize if the question is bit trivial for mathoverflow, but I asked on stack exchange a while ago and haven't got any answer. Let $k$ be a field of characteristic $p > 0$. Consider the ...
12
votes
1answer
330 views

Some basic questions on crystalline cohomology

Let $X_0$ be a smooth projective variety over $\mathbf{F}_q$ and ${X}$ its base change to an algebraic closure $k$ of $\mathbf{F}_q$. Crystalline cohomology $H^*_{\rm cris}(X) := H^*((X/W(k))_{\rm ...
4
votes
1answer
174 views

Motivation for using etale topology in representability of functors problems

I am reading a paper that proves the representability for certain functors whose domain is the category of superschemes. The paper claims that to prove representability of functors (or possibly just ...
2
votes
1answer
105 views

On connectedness of intersection of subgroups

I am quite interested in any partical answer to the following general (maybe a little bit vague) question: Is there some criterion about the connectedness of the intersection of two connected ...
0
votes
0answers
53 views

Quadrics over the univariate function field with discriminant of minimal degree

Consider a non-degenerate quadric $Q(x,y,z) \subset \mathrm{P}^2$ over the univariate function field $\mathbb{F}_p(t)$, where $\mathbb{F}_p$ is a prime finite field, $p > 2$. For simplicity assume ...
17
votes
1answer
508 views

Is being of general type stable under generization

This question is about how varieties of general type over an algebraically closed field of characteristic zero $k$ behave under generization in families. Definition. An integral projective ...
9
votes
0answers
174 views

Quantum Hamiltonian reduction and Quantum Airy structure

I have been reading Kontsevich and Soibelman's "Airy structures and symplectic geometry of topological recursion" (https://arxiv.org/pdf/1701.09137.pdf) and having trouble understanding their Section ...
1
vote
0answers
80 views

How does this $\chi \in X^*(P)$ define a line bundle on $P \backslash G$, where $G$ is a semisimple linear algebraic group

Let $F$ be a number field, and $G$ be a semisimple linear algebraic group over $F$. We let $P_0 \subseteq G$ be a minimal $F$-rational parabolic subgroup. Let $P$ be a standard (i.e. containing $P_0$...
1
vote
1answer
76 views

Equalizer of local analytic isomorphisms

Let $a,b : V\to W$ be two morphisms of smooth complex analytic spaces. Assume $a$ and $b$ are local analytic isomorphisms. Does the equalizer $U$ of $a,b$ exist as a smooth complex analytic ...
4
votes
0answers
191 views

Algebraic Geometry Over the Surreal and Surrcomplex Numbers

I was wondering whether or not there is some kind of theory of algebraic geometry over the field of Surreal and Surrcomplex numbers?
1
vote
0answers
95 views

Resolution of Kleinian Singularities using Hilbert schemes of points

Apologies in advance for the naive and rather speculative question. In this blog post by John Baez, he paints a (perhaps not original) picture of how one might expect that the minimal resolution of a ...
4
votes
0answers
116 views

Does linear independence imply algebraic independence for partitioned homogeneous polynomials?

Define a partitioned homogeneous polynomial of degree $d$ to be a polynomial in $$\mathbb Z[x_{11},\dots,x_{1n},\dots,x_{d1},\dots,x_{dn}]$$ with monomials from entries in (polynomials that are $d$-...
3
votes
1answer
133 views

Effective Cartier divisor is an open property

Let $X$ be a regular affine $\mathbb{C}$-scheme, $A$ a (finitely-generated) $\mathbb{C}$-algebra. Let $Y \subset X \times \mathrm{Spec}(A)$ be a closed subscheme of codimension $1$ such that for each $...
1
vote
0answers
139 views

Continuity of Intersection Pairing on Chow monoids

Let $X$ be a smooth irreducible complex projective variety. As we know, if $\alpha,\beta$ are two cycles intersecting properly in $X$, we can define, via Serre's Intersection Formula, their ...
1
vote
1answer
122 views

Image of smooth curve containing the image of a point as smooth point

Let $f: X \to Y$ be a finite, surjective morphism of smooth, quasi-projective varieties over a field $k$ of characteristic zero. Let $p\in X$. If $\dim X>0$, then does there necessarily exist a ...
2
votes
1answer
91 views

is the induced map of an embedding an Iso on Ext-groups?

I am sorry, but I am quite new to Ext groups of sheaves. However, I have a closed embedding of projective $\mathbb{C}$-schemes $\iota : X \hookrightarrow Y$ and was wondering if $$\iota_*:\mathrm{Ext}^...
2
votes
1answer
102 views

Action of birational map $f$ on the divisor class of line $[H]$

In the paper "Normal Subgroups in the Cremona Group", it is stated that the induced isometry $f_{\ast}$ of $f\in J_d$, where $J_d$ denote the set of Jonquières transformations of degree $d$, satisfies ...
6
votes
1answer
179 views

smoothness of Hurwitz spaces with arbitrary ramification profiles

Fix integers $n\ge 3,d\ge 2$, and partitions $\lambda_1,\ldots,\lambda_n$ of $d$. Let $\mathcal{H}$ be the moduli space of degree $d$ covers $f:C\to\mathbb{P}^1$ that have ramification profiles $\...
4
votes
1answer
78 views

Holomorphic Poisson structures on $C P^{n-1}$ and homogeneous Poisson structures on $C^n$

Is it correct that any holomorphic Poisson structure on $C P^{n-1}$ can be lifted to a homogeneous Poisson structure on $C^n$? By homogeneous I mean a quadratic Poisson structure of the form $\{z_i,...
0
votes
0answers
77 views

Do Plucker relation follow from a subsystem of equations?

The following system of equation: $i, j \in \mathbb{Z}_{\ge 1}$, $i+1<j$, $J \subset \mathbb{Z}_{\ge 1}$, $J \cap \{i,j,i+1,j+1\} = \emptyset$, \begin{align*} P_{i,j,J}P_{i+1,j+1,J} = P_{i,j+1,J} ...
2
votes
1answer
210 views

Is the support of a flat module generically flat?

Let $X$ be an affine, complex variety, $A$ be a $\mathbb{C}$-algebra (not necessarily noetherian) and $F_A$ is a coherent sheaf over $X \times \mbox{Spec}(A)$, flat over $\mbox{Spec}(A)$. Denote by $Y ...
1
vote
1answer
165 views

Leray spectral sequence from hypercohomology

Context: Deligne, Theorie de Hodge II, section 1.4.8. Let $f:X\rightarrow Y$ be a map between spaces; $\mathcal{F}$ a sheaf of abelian groups on $X$, and $\mathcal{F} \rightarrow \mathcal{F}^{\...
1
vote
0answers
172 views

About quotient varieties

Let $K$ be a field, $L/K$ be a finite Galois extension with Galois group $G$ such that $(char(K),|G|)=1$ and $K$ contains all $|G|$th roots of unity. Let $B$ be a $L$-algebra of finite type endowed ...
2
votes
0answers
136 views

Examples of endomorphisms of complex curves

I am looking for examples of smooth projective complex curves $X$ of genus at least $2$ and with algebraic endomorphisms $f:X\rightarrow X$ of degree at least $2$. In the case of elliptic curves and $\...
4
votes
1answer
125 views

$h$ is identity as soon as $h(\Sigma)\cap \Sigma$ contains at least 5 points

In the paper "Normal Subgroups in the Cremona Group", under remark 5.1 they stated that for any generic set $\Sigma \subset \mathbb{P}^2_\mathbb{C}$ of $k$ points, and $h$ is an automorphism of $\...
6
votes
1answer
268 views

How to compute Galois representations from etale cohomology groups of a generalized flag variety?

Let $G$ be a connected reductive group over a number field $K$, $P$ be a parabolic subgroup of $G$ defined over $K$, $X=G/P$ be the generalized flag variety which is a smooth projective variety over $...
9
votes
0answers
246 views

Commuting matrix variety $[A,B]=0$ - can one geometrically explain divisibility of $F_ q$ point count by high powers of $q$?

$\DeclareMathOperator\Comm{Comm}\DeclareMathOperator\Id{Id}$Consider the variety $\Comm$ of commuting matrices $[A,B]=0$ over some field $K$. It is much studied, and interesting for various reasons. ...
7
votes
0answers
114 views

Are maximal tori conjugate Zariski-locally?

Let $S$ be a scheme and let $G\to S$ be a reductive group scheme. Then $G$ admits a maximal torus etale-locally, and any two maximal tori are conjugate etale-locally, by Theorem 3.2.6 and Corollary 3....
3
votes
1answer
221 views

Restriction of Ext sheaves on closed subschemes

Let $f:X\rightarrow C$ be a morphism, where $C$ is a smooth curve. For $t\in C$ let $i_t:X_t = f^{-1}(t)\rightarrow X$ be the inclusion of the fiber of $f$ over $t$, and let $\mathcal{F}$ a coherent ...
4
votes
0answers
89 views

A relation between $Spec((1+I)^{-1}R)$ and $Spec(R/J)$

Let $R$ be a commutative ring with identity and let $I$ and $J$ be two finitely generated ideals of $R$. Clearly $1+I:=\{1+i:i\in I\}$ is a multiplicative closed subset of $R$. We can consider the ...
2
votes
1answer
264 views

Can one always find a bigger global resolution

Let $X$ be a scheme. Let $E$ be a perfect complex of coherent sheaves on $X$ and suppose it admits two global resolutions $ F$ and $F'$. By global resolution I mean that both $F$ and $F'$ are quasi-...
2
votes
0answers
111 views

Restriction of the sheaf of relative differentials

Let $f:X\rightarrow C$ be a morphism, where $C$ is a smooth curve, and let $\Omega_f$ be the sheaf of relative differentials. For $t\in C$ let $i_t:X_t = f^{-1}(t)\rightarrow X$ be the inclusion of ...
3
votes
0answers
73 views

Isogeny to a semi-simple group

Let $G$ be a split reductive group over a field of characteristic zero. Is there always an isogeny $G \to H \times T$ with $H$ semi-simple and $T$ a split torus ? I have in mind the case of $\text{GL}...
10
votes
0answers
205 views

Can the orbit with respect to a rotation on the torus hit an algebraic variety infinitely often?

Question: Does there exist a $d$-tuple $\alpha = (\alpha_1,\dots,\alpha_d) \in \mathbb{R}^d$ (with $1,\alpha_1, \dots,\alpha_d$ linearly independent over $\mathbb{Q}$) and an algebraic variety $V \...
2
votes
0answers
130 views

Structure of the End group scheme of an abelian scheme over ring of integers

Let $O$ be the integer ring of a p-adic field $K$ (finite extension of $\mathbb Q_p$), $\mathscr{A}$ be an abelian scheme over $S=\operatorname{Spec O}$, consider the group endohomorphism scheme of $\...
0
votes
1answer
107 views

About a theorem of Mestre

There is a theorem of Mestre witch states: If $K$ is any field, let $p(x) \in k[x]$ is a monic polynomial with degree $2n$, then there exist polynomials $g(x)$ and $r(x)$ with: 1) $g(x)$ and $r(x)$ ...
3
votes
0answers
119 views

Homotopy colimit description of stacks

Let $F$ be an Artin stack. If $p: X \to F$ is an atlas for $F$, can we express $F$, in the $\infty$-category ${\rm Shv}^{\acute{et}}(k)$ of higher stacks, as a homotopy colimit over the simplicial ...
1
vote
0answers
81 views

Bounding the denominator in the canonical bundle formula

My question concerns with Theorem 3.1 in the paper "A canonical bundle formula" by Fujino and Mori. The theorem claims the following: Suppose $X \to C$ is a fiberation whose general fiber $F$ has ...
6
votes
0answers
126 views

Moduli space of flat connections of Lie group over a 2-torus

We know that the moduli space of SU($N$) flat connections over a torus, is equivalent to a complex projective space $\mathbb{P}^{N-1}$ Namely, $$ M_{\rm flat}=\mathbb{E} / {S}_N = \mathbb{P}^{N-1} $$ ...
0
votes
0answers
113 views

Degree bounds and coefficient size in elimination theory?

Suppose we have polynomials is of form $$h_1(x_1,\dots,x_{dn})-c_1\in\mathbb Z[x_1,\dots,x_{dn}]$$ $$\vdots$$ $$h_{nd}(x_1,\dots,x_{dn})-c_{nd}\in\mathbb Z[x_1,\dots,x_{dn}]$$ where $h_1(x_1,\dots,x_{...
10
votes
1answer
397 views

Functoriality of crystalline cohomology

Let $k$ be a perfect field of characteristic $p>0$, $X$ a smooth projective $k$-variety. Denote by $(X/W_n(k))_{\rm cris}$ the small crystalline site of $X$. Let $f : X\to Y$ be a morphism of ...
3
votes
0answers
127 views

Maximum number of integral roots in degree $d$ polynomial?

Given $f(x_1,\dots,x_n)\in\mathbb Z[x_1,\dots,x_n]$ such that Each coefficient is bound in absolute value by $B$ Degree of each variable in any monomial is bound by $d$ Total degree is $d'$ $f(x_1,\...
6
votes
2answers
271 views

Outer Hodge groups of rationally connected fibrations

I believe the following is true and well known. Theorem (?). Let $X$ and $Y$ be smooth, irreducible, projective varieties over $\mathbb{C}$. Let $$ f\colon X\rightarrow Y $$ be a surjective map with ...
0
votes
0answers
109 views

Final step in Coppersmith?

In the final step in Coppersmith technique we have $n$ polynomials (possibly non-homogeneous) in $\mathbb Z[x_1,\dots,x_m]$ where $m\leq n$ and using elimination theory we extract the common integer ...
2
votes
1answer
120 views

The centralizer of a semisimple element which is not contained in any proper parabolic subgroups

Let $G$ be a connected, reductive group over a field $k$. Let $A_G$ be the split component of $G$. If necessary, assume $k$ is perfect. Let $g \in G(k)$ be a semisimple element. Then the ...