Questions tagged [ag.algebraic-geometry]
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
17,182
questions
3
votes
0answers
61 views
how much information $O_K$ points of a formal scheme over $\mathbb{Z}_p$ contain
assume that $Spf\,A\to Spf\,\mathbb{Z}_p[[t_1,...,t_n]]$ is a closed immersion of flat integral formal schemes over $\mathbb{Z_p}$. I see Kisin several time use that if $Spf\,A(O_K)\subset SPf\,\...
1
vote
0answers
69 views
Unibranch points (definition for varieties over arbitrary field)
In David Mumford's book Algebraic Geometry I, Complex Projective Varieties
treating mainly complex varieties as objects of interest on page
43 he defines what is a topologically unibranch variety $X$ ...
3
votes
0answers
126 views
Is there bijective correspondence among the following $3$ sets?
Let $K \supset \mathbb{Q}_p$ be the $p$-adic field and let $O_K$ be its ring of integers and $M_K$ be the maximal ideal with integral closure $\bar{M}_K$. Consider a power series $f(x) \in O_K[[x]]$ ...
5
votes
0answers
137 views
Serre: Duality of regular differentials
I already asked this on math.stackexchange.com, but didn't get any responses. I hope it is appropriate here.
Let $X'$ be an irreducible singular algebraic curve over an algebraically closed field $k$, ...
3
votes
1answer
92 views
Fibers of Hitchin fibration are equidimensional
Let $X$ be a smooth projective curve over $\mathbb{C}$ of genus $g\ge 3$, $M$ be a moduli space of stable vector bundles on $X$ of rank $n\ge 2$ and degree $d$, $\mathcal{M}$ be a moduli space of ...
2
votes
0answers
36 views
What are possible applications of 'fast arithmetic' in the Jacobian (degree zero Picard group) of projective curves over fields?
It is well known that there are plenty possible applications of 'fast arithmetic' (that is, 1. having an algorithm at hand that actually computes in..., and 2. the running time of that algorithm is ...
1
vote
0answers
68 views
Cotangent bundle of moduli space of stable bundles
I think this is a basic and dumb question.
Let $X$ be a smooth projective curve over $\mathbb{C}$, $M$ be a moduli space of stable bundles and $\mathcal{M}$ be a moduli space of semistable higgs ...
2
votes
0answers
76 views
Variation of Morse Functions: a reference request
Suppose I have a manifold $X$ and a family of Morse functions $F_t:X \times \mathbb R \to \mathbb R$ on it where $t$ is the second parameter. So, if we fix $t$, we get a regular Morse function for ...
2
votes
0answers
54 views
Polarization of Prym varieites
I'm trying to understand polarization and rational Hodge structure of spectral curves and Prym varieties.
Excuse me that this is similar to my previous question.
I want to prove the following,
Let $X$...
8
votes
0answers
146 views
Confusion on basic example of homological projective duality
I've been trying to learn more about Kuznetsov's homological projective duality, and I'm confused about a remark in his notes for his ICM address. At the end of section 4.1, he says:
In particular, ...
1
vote
0answers
102 views
computational complexity of rational parametrisation
Let $\cal C$ be a space curve defined as the common zeros of $n-1$ polynomials. Assuming that $\cal C$ admits a rational parametrisation, there is an algorithm generating a point on the curve through ...
8
votes
0answers
145 views
Leech lattice and rational varieties
Question: Is there a smooth rational variety $X$ of complex dimension $4n$, $n \in \mathbb{N}$; such that the intersection form on $H^{4n}(X,\mathbb{Z})$ is the Leech lattice?
My motivation is mainly ...
6
votes
4answers
188 views
Blow up of projective variety $P^1 \times P^1… \times P^1 (n $ times) and Blow up of $P^n$
It is known that Blow up of $P^1 \times P^1$ at a point is isomorphic to Blow up of $P^2$ at two points. Wondering if there is any general statement for Blow up of $P^1 \times P^1 \times.... \times ...
6
votes
0answers
128 views
nonvanishing higher cohomology of a very ample divisor
I am looking for smooth projective varieties $X$, with $h^i(X, \mathcal{O}_X) = 0$ for $i > 0$, with a very ample line bundle $L$ with some nonvanishing higher cohomology.
What is clear:
(1) Curves ...
4
votes
0answers
124 views
Can we attach (formal) abelian varieties to $p$-adic modular forms?
The Jacobian of the modular curve $X_1(N)$ over $\mathbb Q$ $J_1(N)$ can be decomposed up to isogeny, as a product of abelian subvarieties $A_f$ corresponding to Galois conjugacy classes of Hecke ...
2
votes
0answers
139 views
Is there a proper smooth variety in characteristic $p$ whose Hodge-to-de Rham spectral sequence does not degenerate at $E_1$?
By Deligne-Illusie, such a variety has no lifting to $W_2(k)$. In their paper they state that they do not know if such an example exists. Has this question been answered since then?
5
votes
0answers
87 views
Conjugacy classes of plane k-jet group
Define $G(n, k)$ as a subgroup of $\rm{Aut}(\Bbb C[[x_1, \dots, x_n]]/\mathfrak m^{k+1})$ with identity linear part (so, group of $k$-jets of selfmaps of $\Bbb C^n$). I'm interested in the map from $G(...
6
votes
3answers
421 views
Spectrum of a ring (studied by Krull?) of rational functions
Let $k$ be an algebraically closed field and $\mathbb A^2_k=\operatorname {Spec}k[x,y]$ the affine plane over $k$.
Consider the ring $R \subset k(x,y)$ of the rational functions on the plane defined ...
1
vote
0answers
100 views
The compactified Jacobian is birational to a $\mathbb{P}^1$-fibration over the Jacobian of normalization
Let $Y$ be an integral curve whose only singularity is one simple node at a point $y$, and
$\pi:X\rightarrow Y$ be the normalization with $\pi^{-1}(y)=\{x,z\}$. $J(X)$ is the Jacobian of $X$, and $\...
2
votes
1answer
153 views
Extension of a holomorphic vector bundle on a nodal curve
I am reading a paper on holomorphic curves and stuck in an argument about extension of a given holomorphic vector bundle over a nodal curve.
Let $C$ be a nodal curve without closed componets and $E$ a ...
9
votes
1answer
386 views
What is the automorphism group of the projective line minus $n$ points?
$\DeclareMathOperator{\AGL}{\operatorname{AGL}}\DeclareMathOperator{\PGL}{\operatorname{PGL}}$What is the automorphism group of $\mathbb P^1$ minus $n$ points (let's say over an algebraically closed ...
1
vote
0answers
100 views
Embeddings of Hirzebruch surfaces $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(n))$
Let $X_n=\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(n))$ be the $n-$th Hirzebruch surface. We know that for $d>0$ and higher $k>>0$ the linear system $$\mathcal{L}_{...
3
votes
1answer
94 views
Varieties in the $\mathit{GL}(V)$-module $S_{\lambda}V$
It is known that, given a complex vector space $V$ of dimension $\mathit{dim}(V)=n+1$, all irreducible representations for the group $\mathit{GL}(V)$ are parametrized by Young tableaux $\lambda$.
For ...
0
votes
0answers
57 views
Minimum number of generators of the product of ideals
Let $k$ be an algebraically closed field, let $I, J \subseteq k[x_1,\dots, x_n, y_1,\dots, y_m]$ be ideals, and let $i,j$ be the minimum number of generators of $I$ and $J$, respectively. I have two ...
6
votes
1answer
141 views
Strict transform of a tangent curve under blow-up
$\DeclareMathOperator{\Bl}{\operatorname{Bl}}$It is known that if we have a projective variety $X$ and a projective smooth subvariety $Y$ then the exceptional divisor $E \subset \Bl_{Y}X$ of the blow-...
1
vote
0answers
86 views
Fiber of the Hitchin map
Let $C$ be smooth projective curve, $\mathcal{M}$ be the moduli space of semistable Higgs bundles, $H:\mathcal{M}\rightarrow W= \bigoplus H^0(C,K_{C}^{\otimes i})$ be the Hitchin map, and $\pi :C_s\...
4
votes
0answers
60 views
Generalizations of a theorem of Springer to other varieties
Let $F$ be a complete discretely valued field with ring of valuation $R$, uniformizer $\pi$, and residue characteristic $\neq 2$.
A theorem of Springer says that a quadratic form $q=q_1 \bot \pi q_2$, ...
2
votes
1answer
107 views
References for relative ext-sheaves
I would like to know if there are references for relative ext-sheaves. The only one that I have is the paper of H. Lange (Universal families of extensions, Journal of Algebra 83, 101-112 (1983)), and ...
4
votes
0answers
109 views
Quasi-unipotent monodromy for variation of Landau-Ginzburg cohomology
A pair, $(X,f)$, consisting of a smooth variety and a global function $f:X\rightarrow\mathbb{A}^{1}$ is called a Landau-Ginzburg model, LG-model for short. The LG-cohomology of the pair, dentoed $H(X,...
6
votes
1answer
313 views
Are the Galois actions on automorphisms of twists isomorphic?
This might be a trivial question and I might be overlooking something:
Suppose $k$ is a field with algebraic closure $\overline k$ and absolute Galois group $\Gamma$. Let $X,Y$ be two distinct ...
-5
votes
0answers
33 views
Calculating the circumference of a circle formed from a linear distance [closed]
If you took a linear distance of 10 m and wrapped it around to form a circle would the circumference of the circle be 10 m and ,if not, why not or what would it be? Is there any equation to do this ...
1
vote
0answers
150 views
Almost ring theory and derivations
I don't understand the definition of $\boldsymbol{\Omega}_A$ in the context of almost rings. In Gabber and Ramero https://arxiv.org/pdf/math/0409584.pdf it is covered in 9.6.12. How is $\boldsymbol{\...
6
votes
0answers
184 views
Strict Henselization vs base-change to algebraic closure
Let $x$ be a smooth $k$-point on a variety $X$ over a field $k$ of characteristic $0$.
Is the strict Henselized local ring $\mathcal{O}_{X,x}^{\mathrm{sh}}$ the same as $\mathcal{O}_{X,x}^{\mathrm{h}} ...
4
votes
0answers
86 views
Example of a computation of the volume of a subvariety in projective space $\mathbb{P}^n$
Let us consider the projective space $\mathbb{P}^n$ with the standard Fubini Study metric.
I searched all over the internet but I can't find an example of a calculation of the volume for a projective ...
6
votes
0answers
151 views
Geometric foundation of the Grothendieck polynomials
Grothendieck polynomials were firstly defined in
Alain Lascoux and Marcel-Paul Sch¨utzenberger. Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une vari´et´e de drapeaux....
2
votes
0answers
97 views
The “easier” case in Knudsen's stabilization of pointed curves
My question concerns Knudsen's (super-important) proof that $\overline{M}_{g,n+1}$ is the universal curve over $\overline{M}_{g,n}$ (F. Knudsen, The projectivity of the moduli space of stable curves ...
6
votes
0answers
77 views
Toric description of tautological bundle associated to a weighted projective space
Let $a_1,\ldots,a_n\in\mathbb{N}$ be pairwise coprime positive integers, and consider the weighted projective space $$\mathbb{P}(a_1,\ldots,a_n).$$
It is well known that a weighted projective space is ...
1
vote
0answers
68 views
generalizations of semi-continuity theorems
I would like to know some references (if any) about the generalization of the semicontinuity theorem and "Cohomology and base change" theorem (theorems 12.8 and 12.11 in the book "...
3
votes
0answers
112 views
Cohomology of Siegel modular varieties
$\mathcal{A}_g(N)$ is the moduli space of principally polarized abelian varieties with a level $N$ structure.
Set $C_g=\displaystyle{\lim_{\rightarrow}} H^3(\mathcal{A}_g(N), \mathbb{F}_p)$ where the ...
3
votes
0answers
74 views
Using principal polarisation to “cancel” Jacobian summands in isomorphism
I'm working through the sketch proof of irrationality of cubic threefolds in Huybrechts' The geometry of cubic hypersurfaces.
Let $J(X)$ denote the intermediate Jacobian of a cubic threefold $X \...
4
votes
0answers
162 views
Moduli of finite-dimensional algebras
Let $n\geq 1$ be an integer. There is an obvious family of $n$-dimensional unital algebras parametrized by $\mathbb{C}^{n(n-1)^2}$ such that any $n$-dimensional unital algebra is isomorphic to at ...
4
votes
0answers
71 views
Computing the Hilbert series of an irreducible component of a complete intersection
There's a nice formula for the Hilbert series of any complete intersection of hypersurfaces $X_1\cap\cdots\cap X_i\subseteq\mathbb{P}^n$ in terms of the degrees of $X_1,\ldots,X_i$. Is there a way to ...
6
votes
0answers
313 views
+50
A question about pullbacks of $C^\infty_M$-modules
First, let me state a definition: Let $M$ be a smooth manifold and suppose $\mathcal{E}$ is a sheaf of $C^\infty_M$-modules. Given a point $x \in M$ let $I_x$ denote the vanishing ideal at $x$. We ...
3
votes
0answers
70 views
Cohomology with supports of dualizing sheaf
Let $Z$ be a closed subvariety of $X$ and $T_{Z/X}$ the relative tangent complex. Possibly I want to assume that $Z \hookrightarrow X$ is a regular embedding, so that $T_{Z/X}$ is just a shift of the ...
0
votes
0answers
46 views
Natural correspondence between the set of morphisms and the set of global sections
I'm trying to prove the following claim.
Let $X$ : smooth projective scheme over $\mathbb{C}$, $L$ : line bundle over $X$ and $A$ : $\mathbb{C}$-algebra.
$X_A=X\times_{\mathbb{C}} \operatorname{Spec}A$...
1
vote
1answer
118 views
Divisors on the symmetric product of an elliptic curve
Assume that $C$ is an elliptic curve and $C_p$ is the $p$-fold symmetric product. Let $\beta:C_p\to C$ be defined by the addition on the elliptic curve. Let $u\in C$ be the zero in the additive group ...
0
votes
0answers
31 views
The data of bundle structure of $\operatorname{End}_{0}E$ with Lie algebra
This may be a trivial question. I want to show the following claim about the structure of Lie algebra bundles,
Let $X$ be a smooth projective curve over $\mathbb{C}$, and $E$ be vector bundle over $X$...
15
votes
0answers
195 views
Why should an abelian variety with few places of bad reduction and a lot of endomorphisms not have many points?
In the paper "Points of Order 13 on Elliptic Curves" by Mazur-Tate, they say in the introduction:
It seemed ... that if such an abelian variety $J$, which has bad reduction at only one ...
0
votes
0answers
86 views
The isomorphism classes of lifts of a $\operatorname{PGL}_r$-bundle to $\operatorname{GL}_r$
I want to show the following Lemma,
The set of algebraic isomorphism classes of lifts to $\operatorname{GL}_r$ of an algebraic principal $\operatorname{PGL}_r$ -bundle $P$ on a smooth projective ...
4
votes
0answers
127 views
Perverse restriction
Let $f:U\to V$ be a separable dominant morphism of irreducible positive-dimensional varieties. Let $F$ be a perverse sheaf on $U$. Are there infinitely many closed points $p\in V$ such that $F|_{U_p}$ ...
