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

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96 views

Is the equivariant Gysin map an $H_G^*(\text{pt})$-module morphism?

Let $G$ be a complex reductive group, $X$ a smooth projective variety on which $G$ acts algebraically, and $Y \subseteq X$ a $G$-invariant smooth closed subvariety such that $X\setminus Y$ is also ...
2
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1answer
254 views

Curves of high genus with many rational points

The seminal theorem of Faltings confirms Mordell's conjecture: that is, curves of genus at least 2 have at most finitely many rational points. The proof of Faltings' theorem is not effective, meaning ...
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2answers
254 views

question about Picard scheme

$\underline{Pic}^0(X)$ denotes Picard scheme of $X$. Let $X$ be a nonsingular projective surface over an algebraically closed field $k$ and let $D$ be an effective divisor on $X$. Suppose there ...
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103 views

What is the combinatorial data classifying non-normal affine toric varieties?

Recall that a toric variety is a variety $V$ containing an open dense algebraic torus. Here an algebraic torus means a finite product of copies of the multiplicative group of the ground field (which I ...
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0answers
283 views

Why is this frobenius acting like that frobenius?

My question came up while reading this article by Nicholas Katz, specifically lemma 4.2. I don't think it's necessary to read the article to answer the question, but I'm including it anyways for ...
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1answer
142 views

Complexity of rational $\mathrm{GL}_{n(r)}$-modules

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $G=\mathrm{GL}_n(k)$ for some natural number $n$. For any integer $r\ge 1$, let $G_{(r)}$ denote the $r$th Frobenius ...
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1answer
112 views

Relation between intersection multiplicity and Hilbert-Samuel multiplicities

Suppose $X$, $Y$, $Z$ are projective varieties in $\mathbb{P}^n_K$ of dimension $n-1$, where $K$ is a field. $X$, $Y$, $Z$ intersect properly, and $P$ is one of their intersection irreducible ...
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1answer
228 views

Continuity of l-adic cohomology: is the cohomology of the generic point isomorphic to the completion of the limit of cohomology of open subvarieties?

Let $X$ be a variety over an algebraically closed field $k$. Denote by $\eta$ its generic point; it is the inverse limit of the open subvarieties $X_i$ of $X$. It is well known that the etale ...
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1answer
63 views

reduction of an admissible filtration

Let $(R,m)$ be a local ring and $I$ an $m$-primary ideal of $R.$ $\lbrace I_n\rbrace_{n\in\mathbb{Z} }$ is called $I$-admissible filtration 1) if $m\geq n$ then $I_m\subset I_n.$ 2) for all $m,n,$ ...
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3answers
250 views

Closed orbits of complete flags in $\mathbb{C}^n$

Let $B$ be a symmetric (or antisymmetric) non-degenerate bilinear form on $\mathbb{C}^n$ and let $G$ be the associated group of automorphisms $O(n)$ (resp. $Sp(n)$). What can we say about the ...
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82 views

algebraic varieties whose fundamental group is subgroup separable wrt subvariety subgroups

Call an algebraic variety $\pi_1$-subgroup separable iff, for every $Y\subseteq X$ a closed subvariety and $\hat Y\xrightarrow{i} Y$ a normalisation of $Y$, and subgroup $\Gamma=Im(\pi_1(\hat ...
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1answer
113 views

Properties of extreme rays

Let $X$ be a projective variety over $\mathbb{C}$, then the effective curves module the numerical equivalence form a cone. For my understanding, if the extreme ray $[C]$ has the property that ...
3
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1answer
110 views

Relations between the cohomology of discrete groups and of profinite groups

Let $G$ be a discrete group and $K$ be the profinite completion of $G$. Let $C_K$ denote the category of contionuous $K$-modules and ${C_K}'$ denotes category of finite continuous $K$-modules. Now for ...
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1answer
87 views

local cohomology of Buchsbaum ring

Let $(R,m)$ be a Buchsbaum ring of dimension d. Can we say that $d$-th local cohomology $H_{m}^d(R)$ has finite length?
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84 views

Compute the radical for an ideal without computing the Grobner basis

Given a set of generators for a polynomial ideal $I = <f_1, ..., f_n>$, is it possible to compute a set of generators for the radical $\sqrt I$ without first computing a Grobner basis for $I$?
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0answers
91 views

Algebraic methods in pde [duplicate]

I'm finding myself with a linear, very symetric system of first orders pde with polynomial coefficients. Wandering on the web, i learnt there is some nice alebraic way to deal with it involving ...
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1answer
114 views

Good software for solving a system of algebraic equations [closed]

I need to solve the following system of equations. solve([ a*c*e=a2*c2*e2, b*d*f=t*b2*d2*f2, t*a1*c1*e1=a3*c3*e3, b1*d1*f1=b3*d3*f3, a*b*a1*b1=a2*b2*a3*b3, c*d*c1*d1=c2*d2*c3*d3, ...
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1answer
681 views

What is the meaning of $(h^{11},h^{21})\to (h^{11}-240,h^{21}+240)$ in Calabi-Yau threefolds?

By browsing through the Hodge data of known Calabi-Yau threefolds, I stumbled upon an observation that frequently enough a pair of Hodge numbers $(h^{11},h^{21})$ comes together with the pair $ ...
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1answer
82 views

Relation between local cohomology and koszul cohomology of multigraded ring

Let $R$ be a ${\mathbb {Z}}^k$-graded Noetherian ring, $J=(x,y)$ an ideal of $R$ where $x,y\in R_{(1,\ldots,1)}.$ Is this following true $$H_{J}^i(R)=\underset{n}\varinjlim{H^i((x^n,y^n),R)},$$ where ...
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1answer
280 views

degree 7 rational curves through ten points in P4

This is a very classical flavoured question, and probabaly it is not difficult. I would like to know the shape of the space of rational degree 7 curves in $P^4$ that pass through 10 fixed points. By ...
2
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1answer
128 views

mixed Hodge structure on Cohomology with compact support

Let $X$ be a smooth quasi-projective variety of dimension $n$ over the complex numbers (let us assume it is the complement of a normal crossings divisor in a smooth projective variety). Then how does ...
5
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1answer
569 views

A naive algebraic geometry question

Suppose $X$ is a scheme over a ring $A$, $B$ is an $A$-algebra, and $X\times_AB$ is affine. I am looking for conditions on $A$ and $B$ (and perhaps the structure morphism of $X$ over $A$) that will ...
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0answers
95 views

Logarithmic de Rham complex induced map on Hypercohomology

• Question: Assume X is an algebraic manifold over $\mathbb{C}$ and $D$ is a simple normal crossings divisor. The logarithmic de Rham complex $\left(\Omega_{X}^{\bullet}\left(log\ ...
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2answers
91 views

Divisors with positive Iitaka dimension

Let $X$ be a non-singular projective variety, and $D$ a divisor on $X$. Saying that $D$ has positive (meaning non-zero) Iitaka dimension is equivalent to the function $n \mapsto h^0(\cal{O}(D))$ ...
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2answers
125 views

Dimension of the nilpotent centralizer of a nilpotent matrix

Fix a natural number $n$ and an algebraically closed field $k$. Let $\mathfrak{g}=\mathfrak{gl}_n(k)$. For any partition of $n$, $\lambda=(\lambda_1,\ldots,\lambda_r)$, let $A_{\lambda}$ be the ...
4
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0answers
206 views

Spectral sequences and Koszul complexes in Deformation Theory

I'm reading this paper of A. Vistoli and I have some questions about the discussion in page 5. This is the context (If you don't want to download the paper): Let $A'$ be a noetherian local ring with ...
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0answers
45 views

Bounding a sum of functions defined on effective divisors

The following is from a PhD thesis "Algebraic Circle Method" by Thibaut Pugin, which I am currently reading. Let $k$ be a finite field of order $q$. Let $X \subseteq \mathbb{A}^{n+1}_k$ be the ...
6
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2answers
287 views

When are two subvarieties of matrices conjugate?

Let $X$ and $Y$ be two subvarieties of $n\times n$ matrices. My question is that is there any condition to guarantee that there exits some matrix $g$ such that $Y=g^{-1} X g$? If such $g$ exists, then ...
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0answers
114 views

Ratio of periods for elliptic curves in an isogeny class

Let $E \to E^\prime $ be an isogeny of elliptic curves defined over $\mathbb{Q}$. Then what is the definition of ratio of periods for elliptic curves in the isogeny class and how to calculate the ...
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2answers
222 views

The Schottky group and the fundamental group of a compact Riemann surface

I am quoting the following description from a paper, "...every compact Riemann surface can be obtained as the quotient $\mathbb{C}/\Gamma$ where $\Gamma$ is a Schottky group. The Schottky group of a ...
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0answers
98 views

when is an irreducible SL_2(C) representation of a cusped hyperbolic 3-manifold scheme reduced or smooth

Let $M$ be an orientable cusped hyperbolic 3-manifold. Let $\rho \in hom(\pi_1(M),SL_2(\mathbb{C}))$ be an irreducible representation. Is $\rho$ scheme reduced ? What can prevent $\rho$ from being ...
1
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1answer
138 views

Picard number of Prym Variety

It is known that, for a generic curve $C$, $End(J(C)) \cong \mathbb{Z}$, hence picard number is one. Is this true for Prym Variety?
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0answers
113 views

Question related to the number of rational curves on a hypersurface

The following is from a PhD thesis "Algebraic Circle Method" by Thibaut Pugin, which I am currently reading. Let $k$ be a finite field of order $q$. Let $\mathfrak{X} \subseteq \mathbb{P}^n_k$ be ...
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2answers
334 views

Is the category of schemes wellpowered? regularly wellpowered?

Wellpowered means that for every scheme $X$, the subobject lattice of monormophisms $Y \to X$ is essentially small; regularly wellpowered means that for every scheme $X$, the regular subobject lattice ...
1
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1answer
223 views

Singular cohomology as a Zariski sheaf

Let $X$ be a complex algebraic variety, and consider the presheaf $U \mapsto H^i(U^{an}, \mathbb Z)$ in the Zariski topology. Is there a theorem that says this presheaf is already a sheaf, for ...
2
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2answers
158 views

An ample line bundle on a K3 surface

Let $X$ be a K3 surface obtained as a double covering of $\mathbb{P}^1 \times \mathbb{P}^1$ branching along a $(4,4)$-divisor. I think the natural line bundle $\pi^*\mathcal{O}_{\mathbb{P}^1\times ...
4
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0answers
118 views

Continuity of the Hilbert pairing

I would like to know if the Kummer pairing (or the analogue of the Hilbert Symbol) for a one dimensional group defined over the ring of integers of a higher-dimensional local field is continuous (with ...
16
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1answer
575 views

The classical number $2875$ of lines on the quintic, as a DT invariant

On a general quintic threefold $Y\subset \mathbb P^4$ there are $2875$ lines. The result is classical and one can obtain it via a Chern class computation. But $Y$ is a Calabi-Yau threefold, thus one ...
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1answer
254 views

Lie algebraic Grassmannian

Assume that $L$ is a Lie algebra structure on $\mathbb{R}^{n}$, and $1<k<n$ is given. We define $Gr(k,n)_{L}$, the space of all $k$ dimensional Lie subalgebra of $(\mathbb{R}^{n}, L)$. For ...
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1answer
106 views

Genus in weighted projective space

I am reading a paper by Darmon and Granville where they made the following claim, with respect to the generalized Fermat curve $Ax^p + By^q = Cz^r$. They said: "... this (referring to the generalized ...
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1answer
271 views

Bertini's Theorem

Let $p_1,...,p_n\in\mathbb{P}^{N}$ be general points. Consider the linear system $|L|$ of hypersurfaces of degree $d$ in $\mathbb{P}^{N}$ with prescribed multiplicities $m_1,...,m_n$ at $p_1,...,p_n$. ...
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1answer
121 views

Kawamata-Log-Terminal pairs

Let $p_1,...,p_n\in\mathbb{P}^3$ be general points, and let $\Delta\subset\mathbb{P}^3$ be a general surface of degree $d$ with points of multiplicity $m_i$ at $p_i$ for $i = 1,...,n$. Consider the ...
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1answer
198 views

Can one prove that toric varieties are Cohen-Macaulay by finding a regular sequence?

It's well-known that if $P$ is a finitely generated saturated submonoid of $\mathbb{Z}^n$, then the monoid algebra $k[P]$ is Cohen-Macaulay (at the maximal ideal $m = (P-0)k[P]$). However, all proofs ...
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1answer
136 views

Curve of degree $d$ through $2d+1$ points in $\mathbb P^3$

It is known that a Hilbert scheme of degree $d$ curves in $\mathbb P^3$ can have dimension more than $4d$. But, does it imply that for some types of curves there are such a curve through any, say, ...
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68 views

Splitting for Subsequence of Automorphism Sequence for Algebraic Groups

Let $G$ be a split reductive algebraic group over an arbitrary field $k$ Suppose we have a split maximal torus $T$. There is a short exact sequence of groups $$ 1\to \mathrm{Inn}(G)\to ...
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0answers
102 views

Parabolic bundle and chern class

I'm a physicist and I'm trying to figure out some things about parabolic bundles. In particular I'd like to understand which is the relationship, if present, between the parabolic degree and the first ...
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1answer
120 views

Kleiman's and Nakai-Moishezon's ampleness criteria

I would like to work out a simple example to understand the relation between Kleiman ampleness criterion and Nakai-Moishezon ampleness criterion. Namely, let $X$ be the blow-up of $\mathbb{P}^{2}$ at ...
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3answers
341 views

Do any Stone-like dualities have some self-dualities hidden inside them?

This question originated from the observation that in most cases when one has duality of structured sets induced by a dualizing set-with-two-structures $D$, both sides of the duality are substructures ...
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0answers
129 views

Semicontinuity and Grauert's theorems for \'{e}tale coherent sheaves

Let $f:X\to Y$ be a proper morphism, and $\mathcal{F}$ be a coherent sheaf on the small \'{e}tale site $X_{et}$, flat over $Y$. For each geometric point $\bar{y}\to Y$, regard the pull back ...
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3answers
659 views

Introductory References for Geometric Representation Theory

Would anyone be able to recommend text books that give an introduction to Geometric Representation Theory and survey papers that give an outline of the work that has been done in the field? I'm ...