Questions tagged [ag.algebraic-geometry]
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
21,606
questions
3
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Decompositions from torus actions and compactness of (sub-)level sets
Let $T=\mathbb{C}^{*}$ act on a smooth complex quasi-projective variety $X$. Assume that the limit point $\lim_{t\to 0}t\cdot x$ exists for every $x\in X$.
From the induced $U(1)$-action and its (...
- 22.7k
4
votes
0
answers
329
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Is complete intersection a open or closed property in Hilbert schemes
Fix an integer $N$, $X$ a (smooth) complete intersection subvariety in $\mathbb{P}^N$. Denote by $P$ the Hilbert polynomial of $X$ (as a subvariety in $\mathbb{P}^N$). Consider the Hilbert scheme $\...
- 2,126
26
votes
1
answer
798
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What are the points of simple algebraic groups over extensions of $\mathbb{F}_1$?
The "field with one element" $\mathbb{F}_1$ is, of course, a very speculative object. Nevertheless, some things about it seem to be generally agreed, even if the theory underpinning them is not; in ...
- 30.1k
8
votes
1
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A direct proof of a property of symmetric 2x2-determinants
Let $f(a,b,c)=\det\begin{pmatrix}a &b\\ b& c\end{pmatrix}\in\mathbb{R}[a,b,c]$ be the determinant of a $2 \times 2$ real symmetric matrix.
Let $f(x_i,y_i,z_i)\geq 0$, $x_i\geq 0$, $z_i\geq 0$ ...
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1
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0
answers
183
views
Puiseux's theorem's converse
Puiseux's theorem asserts that
given a polynomial equation $P(x,y)=0$, its solutions in $y$, viewed as functions of $x$, may be expanded as Puiseux series that are convergent in some neighbourhood of ...
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4
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1
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compact objects and derived categories
Sorry in advance if my question does not have the required level.
Let $K$ be a commutative ring (let say an integral domain for simplicity) and let $0\neq s\in K$. Let $K[s^{-1}]$ be the localized ...
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1
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Resolving $\mathbb Z_n$ action on $\mathbb C^2$
Consider a diagonal action of $\mathbb Z_n$ on $\mathbb C^2$ generated by $(z_1,z_2)\to (\mu^pz_1,\mu^qz_2)$, with $\mu^n=1$.
Question. Is it always possible to find a smooth blow up $X\to \mathbb ...
- 14k
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0
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209
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Description of flop as graded algebra
I am looking for an example of a flop $Y \to X \leftarrow W$, possibly with exceptional locus at least a $\mathbb{P}^2$, where $X = \text{Spec } A$ is affine and $Y,W$ can be described as explicit ...
- 1,869
9
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198
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Etale maps and local intersection cohomology
Suppose that $f:(X,x) \to (Y,y)$ is etale at $x$, meaning that it induces an isomorphism $C_xX \to C_yY$ on tangent cones. Then $f$ induces an isomorphism from the cohomology of $IC_{X,x}$ (the stalk ...
- 2,527
32
votes
2
answers
1k
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Analogies supporting heuristic: Weyl groups = algebraic groups over field with one element?
There is well-known heuristic that Weyl groups are reductive algebraic groups over "field with one element".
Probably the best known analogy supporting that heuristic is the limit $q\to1$ ...
10
votes
1
answer
732
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Equivariant Riemann-Hurwitz
The Riemann-Hurwitz formula starts with a genus $g$ algebraic curve $Y$ and a ramified cover $\pi\colon X\to Y$ of degree $N$, with ramification indices $e_P$ and computes invariants of $X$, such as ...
- 8,657
3
votes
3
answers
316
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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 ...
user111251
3
votes
1
answer
481
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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 ...
- 7,161
4
votes
0
answers
140
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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 $\{...
- 1,022
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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 ...
- 1,856
11
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0
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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, ...
- 3,519
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0
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272
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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 ...
- 845
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votes
1
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490
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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 $k$-...
- 3,432
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 $X(c)=H^{-1}(c)/S^1$ ...
- 14k
6
votes
1
answer
455
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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 ...
- 3,362
9
votes
1
answer
495
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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 ...
- 65.1k
8
votes
1
answer
281
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 ...
- 113
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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 ...
- 1,135
3
votes
0
answers
595
views
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 ...
- 113
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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 \...
- 1,801
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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 $$G((z))=\coprod\...
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votes
2
answers
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How to define the intersection multiplicity of a projective variety and a complete intersection?
In the appendix of Algebaric Geometry by Hartshorne, he shows us that how Serre defines the intersection number in a more general case:$$i(X,Y;Z)=\sum(-1)^i\cdot\bigl(\operatorname{length} \...
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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,...
- 713
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0
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594
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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 $G$-...
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0
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242
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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 ...
- 1,135
6
votes
0
answers
368
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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 ...
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2
votes
1
answer
293
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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 ...
- 21.1k
13
votes
1
answer
973
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 ...
- 433
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votes
1
answer
2k
views
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 ...
- 17.5k
11
votes
1
answer
514
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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 ...
- 17.4k
1
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0
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543
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 ...
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174
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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 ...
- 10k
6
votes
0
answers
257
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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 ...
user21574
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 $...
- 443
6
votes
1
answer
504
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 ...
- 323
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513
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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 ...
- 411
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0
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366
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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 (...
- 169
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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....
- 22.6k
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?
...
- 51
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 '...
- 59
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 $...
- 1,825
3
votes
1
answer
190
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
703
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 ...
- 505
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 ...
- 13.6k