Questions tagged [ag.algebraic-geometry]
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
21,545
questions
3
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3
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Voronoi and Delaunay
Please provide some references on Voronoi and Delaunay decompositions which is mathematically written. I mean I can find several texts or links on this written for computer science students without ...
3
votes
1
answer
478
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Is $\mathcal{Ext}^i(F,G)$ the sheafification of $Ext^i(F,G)$?
Disclaimer: This was first asked here on math.stackexchange with no answers.
Let $F,G$ be quasicoherent sheaves of modules on a scheme $X$, then is the sheaf $\mathcal{Ext}^i(F,G)$ equal to the ...
4
votes
0
answers
139
views
Uniformization of Riemann surfaces by iso-classical Schottky groups
Let $\Gamma=<g_1, \dots, g_n>$ $\subset PGL_2(\mathbb{C})$ be a Schottky group of rank $n$. The group $\Gamma$ is called classical if there exists a set of $2n$ pairwise disjoint closed balls $\{...
0
votes
0
answers
118
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Extending isomorphism on closed subschemes to open neighborhoods
Let $X, Y$ be schemes, $Z, Z'$ closed subschemes of $X, Y$ respectively. Assume there is an isomorphism $f: Z\to Z'$, is it possible to find open neighborhoods $U, V$ of $Z, Z'$ respectively such that ...
11
votes
0
answers
276
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Generalization of Dwork's Derivation of the Picard-Fuchs equation
Background:
Let $V_\lambda$ be the elliptic curve $x^3+y^3+z^3 - \lambda xyz=0$. Then, when we consider $\omega_\lambda \in H^1_{dR}(V_\lambda)$, since $H^1_{dR}(V_\lambda)$ is only 2-dimensional, ...
5
votes
0
answers
270
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On toroidal compactifications of Hilbert Kuga-Sato varieties
Let $F$ be a totally real field of degree d. There are Hilbert modular varieties over $\mathbb{Q}$ that paramatrize abelian varieties of dimension d with an action of $\mathcal{O}_F$ the ring of ...
3
votes
1
answer
488
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characterisation of regular morphisms
Recall that a morphism of schemes $X\rightarrow Y$ is regular if it's flat with geometric fibres that are regular schemes.
Fix a field $k$ and consider a morphism $f:X\rightarrow Y$ of noetherian $k$-...
7
votes
1
answer
253
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Bimeromorphic equivalence of reduced spaces for Kähler $S^1$-actions
Let $(X,\omega)$ be a smooth Kähler manifold (not necessarily compact) with an isometric $S^1$-action with a Hamiltonian $H$. It is a well known fact that
1) The reduced spaces $X(c)=H^{-1}(c)/S^1$ ...
6
votes
1
answer
453
views
Picard number of a general fiber of a fiber contraction
Suppose in the last step of a MMP, we obtain a Mori fiber space $f: X \to Z$, and let $F$ be a general fiber of $f$, then is the Picard number $\rho(F)$ of $F$ equal to $1$? Notice that the relative ...
9
votes
1
answer
494
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Pull-back of an irreducible ample divisor via an isogeny of abelian varieties
In the (wonderful) book by C. Birkenhake and H. Lange Complex Abelian Varieties we can find the following result, see Corollary 4.3.4 page 77. It is stated in any dimension $g \geq 2$, but let us ...
8
votes
1
answer
280
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Do complex varieties have a dense open subset with residually finite fundamental group?
Let $S$ be a smooth connected variety over the complex numbers. The fundamental group might not be residually finite (i.e., the homomorphism $\pi_1(S(\mathbb C)) \to \pi_1^{\mathrm{et}}(S)$ might not ...
2
votes
0
answers
118
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Is there a hyperkaehler manifold whose mirror is the total space of a tangent/cotangent bundle?
I am looking for an example of a hyperkaehler manifold $Y$ whose mirror is the total space of a tangent bundle $TX$ or a cotangent bundle $T^*X$, where $X$ can be any Riemannian manifold.
Is such a ...
3
votes
0
answers
595
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Monodromy representations are "quasi-unipotent"
Let $S$ be a smooth complex algebraic variety, let $b$ be a closed point of $X$, and let $f:X\to S$ be a polarized family of smooth projective varieties over $S$. This induces a monodromy ...
4
votes
0
answers
112
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How to describe the subspace of invariants under the Rosati involution?
Consider the Jacobian $J_C$ of hyperelliptic curve
$$C\!: y^2 = x^5 + a$$
over a finite field $\mathbb{F}_p$, where $a \in \mathbb{F}_p^*$, $p \equiv 2 \ (\mathrm{mod} \ 5)$, $p > 2$. Let $\pi \...
3
votes
0
answers
138
views
Cartan decomposition for $G[z]$
Let $G$ be a reductive group over complex numbers. Fix some maximal torus $T$. Let $\Lambda^{+}$ be the monoid of dominant coweights. It is known that one has a Cartan decomposition $$G((z))=\coprod\...
4
votes
2
answers
1k
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How to define the intersection multiplicity of a projective variety and a complete intersection?
In the appendix of Algebaric Geometry by Hartshorne, he shows us that how Serre defines the intersection number in a more general case:$$i(X,Y;Z)=\sum(-1)^i\cdot\bigl(\operatorname{length} \...
5
votes
1
answer
246
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The components of the space of projectively flat bundles over a Riemann surface
Recently, I'm reading the paper "Analytic structures on the space of flat vector bundles over a compact Riemann surface" by Gunning. In the introduction of this paper, Gunning says that the set $H^1(M,...
7
votes
0
answers
590
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Automorphisms of semistable $G$-bundles
Let $X$ be a projective smooth curve over an algebraically closed field $\mathbb k$ of any characteristic. Let also $G$ be a reductive group over $\mathbb k$. (Probably) following Ramanathan, a $G$-...
2
votes
0
answers
241
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Is the mirror of a noncompact hyperkaehler manifold also hyperkaehler?
This is essentially a follow-up question from 'Is the mirror of a hyperkaehler manifold always a hyperkaehler manifold?'. Verbitsky's theorem in (https://arxiv.org/pdf/hep-th/9512195.pdf) says that ...
6
votes
0
answers
366
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Faithfully flat descent of projectivity for non-commutative rings
I am looking for a reference for the following statement (or another one explained further below):
Let $M$ be a module over a (not necessarily commutative) ring $R$ and $R'\supset R$ a faithfully ...
3
votes
1
answer
291
views
Example of a morphism of complex spaces or "nice schemes" that is not cohomologically flat in any point
Suppose that $f:X\rightarrow S$ is a proper, separated morphism of complex spaces (with $S$ reduced) and $\mathcal{F}$ a is $f$-flat coherent sheaf on $X$.
From (well-)known results it is known that ...
2
votes
1
answer
288
views
Basic question on dimension of intersection of subschemes
Let $X$ be a Noetherian irreducible scheme of dimension $n$. Let $Y,Z$ be its closed irreducible subschemes of dimensions $k,l$ respectively.
Under what technical conditions the dimension of each ...
13
votes
1
answer
957
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How is a Stack the generalisation of a sheaf from a 2-category point of view?
A stack is usually given in terms of:
-A category $F$ fibered over another $C$ such that the functor $Hom(x,y), x,y \in F(\alpha), \alpha \in C$ is a sheaf
-The descent data are effective.
There ...
35
votes
1
answer
2k
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Are there topological versions of the idea of divisor?
I am trying to extract a particular, more lightweight and more focussed at the same time, case of my recent question Which of the physics dualities are closest in essence to the Spanier-Whitehead ...
11
votes
1
answer
513
views
Oesterlé's unpublished bound on Uniform Boundedness
The bound in Merel's solution to the Uniform Boundedness conjecture is not explicit, as it relies on Falting's work on the Mordell conjecture. I think this still is the case.
But there are known ...
1
vote
0
answers
534
views
Orbifold line bundles
I was going through the paper - http://repository.ias.ac.in/3652/1/427.pdf, and I got this question. Let $Y$ be a connected smooth projective variety of dimension $n$ over $\mathbb{C}$. Let $G$ be a ...
4
votes
0
answers
173
views
Explicit examples of finite unramified group schemes
What are some explicit examples (e.g., by explicitly describing its Hopf algebra) of finite unramified group schemes? (Ie, the sort of group schemes which appear as automorphism groups of objects ...
6
votes
0
answers
256
views
Analytical point of view of Kawamata's Unipotent reduction condition for Calabi-Yau family
Motivation: Unipotent reduction condition is very important for study of family of algebraic varieties. For example for algebraic fiber space if we have such condition then the direct image of ...
4
votes
2
answers
327
views
estimate for a sum of products of Weil's sum
Let $p$ be a prime and consider the field $\mathbb{F}_p$. Fix $f\in\mathbb{F}_p[X]$ a polynomial of degree $d\ge 2$. Define
$$
K(x,y)=\frac{1}{\sqrt{p}}\sum_{z\in\mathbb{F}_p}e_p(xz+yf(z)),
$$
where $...
6
votes
1
answer
503
views
Rationality of the Tate module of an abelian variety relative to the algebra of its endomorphisms
Suppose that $K/\mathbb{Q}_p$ is a finite extension and $k_K$ the residue field of $K$. Let $A/K$ be an abelian variety with good reduction. Suppose that $E\to\mathrm{End}^0_K(A)$ is an inclusion of a ...
5
votes
0
answers
508
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Reduction of torsion points on Neron Model
Let $K/\mathbb{Q}_p$ be a finite extension with ring of integers $R$ and residue field $k$. Let $A/K$ be an abelian variety with Neron model $\mathcal{A}/R$. We denote by $\tilde{\mathcal{A}}/k$ the ...
1
vote
0
answers
363
views
Kawamata covering lemma - question on the branch divisor
Let $X$ be a non-singular projective variety over the field of complex numbers, of dimension $\geq 2$. Suppose $D$ is a non-singular and irreducible divisor of $X$.
The Kawamata covering lemma (...
1
vote
0
answers
201
views
Galois Cohomology and $\sqrt{k} \notin \mathbb{Q}$
I know it seems excessive, but I have been trying to understand the relationship between two concepts:
Galois cohomology
Fermat Descent
The first one is very abstract and I know very little about it....
5
votes
1
answer
208
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When do Gorenstein Stanley-Reisner rings have Du Bois singularities?
The question is pretty much as in the title. Given a simplicial complex $\Delta$, I can associate a Stanley-Reisner ring. I assume this ring is Gorenstein, when does it have Du Bois singularities?
...
2
votes
1
answer
112
views
Smoothness of space of morphisms from a curve to a locally complete intersection
Let $C$ be an irreducible smooth project curve over $\mathbb C$ and $Y$ a variety over $\mathbb C$ locally of complete intersection. Write $Y^{\text sm}$ for the smooth locus of $Y$. Consider the '...
4
votes
2
answers
143
views
Covering all except one of the purple intersection points of $n$ red and $m$ blue lines efficiently
Consider a set of $n$ red lines and $m$ blue lines, suppose there are $nm$ distinct red-blue intersections.
What is the minimum number of lines $L_1,L_2,\dots, L_n$ such that the union contains all $...
3
votes
1
answer
189
views
"Künneth bigrading" for subsets of $X \times Y$?
Given two algebraic varieties $X$ and $Y$, the Künneth theorem implies that there is a relation between $H^*(X) \otimes H^*(Y)$ and $H^*(X \times Y)$, and in fact in many cases they are equal.
Given ...
5
votes
2
answers
697
views
Does the degree of a finite dominant morphism bound the induced degree on subschemes?
Suppose $f: \widetilde{X} \to X$ is a finite dominant morphism between connected, normal, Noetherian schemes, and that this morphism induces a dominant morphism $f_W: \widetilde{W} \to W$ between ...
4
votes
1
answer
215
views
Action of $N(H)/H$ on the colors of a spherical homogeneous space $G/H$
Let $G$ be a semisimple group over $\mathbb C$, and $X=G/H$ be a spherical homogeneous space of $G$.
Let $T\subset B\subset G$ be a maximal torus and a Borel subgroup.
Let $S=S(G,T,B)$ denote the ...
1
vote
0
answers
884
views
Trivial normal bundle
I would like to know if there is a theorem along those lines: let $V$ be a submanifold in $\mathbb{R}^n$ such that $V$ is the boundary of a submanifold with boundary $W$. Then, the normal bundle of $V$...
1
vote
0
answers
311
views
Fixed point set of torus action is discrete and infinite?
Let $T=(\mathbb{C}^*)^k$ act holomorphically on a smooth quasi-projective complex algebraic variety $M$.
Can the fixed point set $M^T$ be discrete and infinite?
I think the answer is no because ...
3
votes
0
answers
512
views
An Explicit Example of Galois Theory for Schemes
I'm currently attempting to understand Galois theory for schemes, largely following the books Galois Theory for Schemes by Henrik Lenstra and Galois Groups and Fundamental Groups by Tamas Szamuely. ...
5
votes
0
answers
268
views
Equivariant cohomology of grassmannian
The Grassmannian $X=Gr(2,\mathbb C^4)$ has an involution $\sigma:V \mapsto V^{\perp}$ with respect to the symplectic form $antidiag(1,1,-1,-1)$. The torus equivariant cohomology of the grassmannian ...
5
votes
1
answer
437
views
Reference for normalization of an algebraic stack?
Is there a standard reference on stacks which discusses (relative) normalization?
This older question seemed to link to someone's notes, but the link is now broken. In any case, it would be nice to ...
2
votes
0
answers
89
views
A question on Lusztig's Intersection cohomology complexes on a reductive group
This is a proposition in Lusztig's Intersection cohomology complexes on a reductive group (Invent. Math. 75, 205-272).
Proposition 6.3. The triple $(L,C_1, \mathcal{E}^{\cdot}_1)$ above is unqiuely ...
0
votes
0
answers
280
views
For which monic irreducible $f \in \mathbb{C}[x,y][T]$, $\mathbb{C}[x,y][T]/(f)$ is a UFD?
Let $f=f(T) \in \mathbb{C}[x,y][T]$ be a monic irreducible polynomial:
$f=T^n+a_{n-1}T^{n-1}+\cdots+a_1T+a_0$,
$a_j \in \mathbb{C}[x,y]$, $0 \leq j \leq n-1$.
Denote $B=\mathbb{C}[x,y,T]/(f)=\mathbb{C}...
3
votes
1
answer
265
views
Semi-stability of Ulrich bundle
A vector bundle $E$ on a smooth projective variety $X$ is called Ulrich bundle if it is Arithmetically Cohen-Macaulay , i.e., $H^i(E(t)) = 0 $ for all $t \in Z$ and $0 < i < k$ and with Hilbert ...
6
votes
1
answer
677
views
Thick subcategories
I hope this question is not too trivial for mathoverfolw.
Let $R$ be a commutative ring (with $0\neq 1$) and $D_{Perf}(R)$ the triangulated category of perfect complexes. Let $C$ be a thick ...
4
votes
1
answer
694
views
Monodromy representation of elementary simple covers
Let $X$, $Y$ be smooth, connected, compact manifolds (for instance, projective varieties) and $f \colon X \longrightarrow Y$ be a finite, branched cover of degree $n$, with branch locus $B \subset Y$. ...
2
votes
1
answer
252
views
Polynomials and matrices in $\Bbb F_q$
Given a polynomial $p(x,y)\in\Bbb F_q[x,y]$ of $(x,y)$ degree $(n_x,n_y)$ ($n_x,n_y\geq0$ and $n_x,n_y\in\Bbb Z$) where $q=p^\alpha$ with $p$ a prime and $\alpha\in\Bbb N$ how many different matrices $...