Questions tagged [ag.algebraic-geometry]

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

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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 ...
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3 votes
1 answer
478 views

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 ...
stupid_question_bot's user avatar
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 $\{...
Daniele Turchetti's user avatar
0 votes
0 answers
118 views

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 ...
Lao-tzu's user avatar
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11 votes
0 answers
276 views

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, ...
Catherine Ray's user avatar
5 votes
0 answers
270 views

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 ...
Bear's user avatar
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3 votes
1 answer
488 views

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 $k$-...
prochet's user avatar
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7 votes
1 answer
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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 $X(c)=H^{-1}(c)/S^1$ ...
aglearner's user avatar
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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 ...
Li Yutong's user avatar
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9 votes
1 answer
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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 ...
Francesco Polizzi's user avatar
8 votes
1 answer
280 views

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 ...
Randy's user avatar
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2 votes
0 answers
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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 ...
Mtheorist's user avatar
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3 votes
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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 ...
Randy's user avatar
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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 \...
Dimitri Koshelev's user avatar
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\...
Tatyana's user avatar
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4 votes
2 answers
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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} \...
Kai's user avatar
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5 votes
1 answer
246 views

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,...
swalker's user avatar
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7 votes
0 answers
590 views

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 $G$-...
user42024's user avatar
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2 votes
0 answers
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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 ...
Mtheorist's user avatar
  • 1,135
6 votes
0 answers
366 views

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 ...
Fabian Meumertzheim's user avatar
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 ...
Horstenson's user avatar
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 ...
asv's user avatar
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13 votes
1 answer
957 views

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 ...
HaroldF's user avatar
  • 433
35 votes
1 answer
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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 ...
მამუკა ჯიბლაძე's user avatar
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 ...
Myshkin's user avatar
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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 ...
gradstudent's user avatar
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 ...
Will Chen's user avatar
  • 10k
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 ...
user avatar
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 $...
Tony B's user avatar
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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 ...
Lukas's user avatar
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5 votes
0 answers
508 views

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 ...
Jędrzej Garnek's user avatar
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 (...
user349424's user avatar
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....
john mangual's user avatar
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5 votes
1 answer
208 views

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? ...
user110984's user avatar
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 '...
user312073's user avatar
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 $...
Gorka's user avatar
  • 1,825
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 ...
W. Cadegan-Schlieper's user avatar
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 ...
Eric Canton's user avatar
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 ...
Mikhail Borovoi's user avatar
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 $V$...
Melchior's user avatar
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 ...
HLC's user avatar
  • 287
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. ...
user avatar
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 ...
Mark's user avatar
  • 185
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 ...
Will Chen's user avatar
  • 10k
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 ...
AzumaTokaku's user avatar
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}...
user237522's user avatar
  • 2,783
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 ...
Simon's user avatar
  • 33
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 ...
M.O.'s user avatar
  • 125
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 $B \subset Y$. ...
Francesco Polizzi's user avatar
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 $...
Turbo's user avatar
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