Questions tagged [ag.algebraic-geometry]

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

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Pull back of D-modules and Koszul resolution

Consider an holonomic D-module on a smooth algebraic variety $X$ over a field $k$ of caracteristic 0. Let $i: Y \hookrightarrow X$ be a regular embedding. $Li^* M = \mathcal{D}_{Y\to X} \otimes^L_{...
AFK's user avatar
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2 votes
2 answers
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Covering seifert manifolds

Let $M$ be a 3-manifold with boundary. If $M$ has an orientable finite cover that is a Seifert fiber space, then is $M$ also a Seifert fiber space?
jhoel's user avatar
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Tangent bundles and birational morphisms

Let $f:X\rightarrow Y$ be a smooth morphism between smooth schemes. Then there is an exact sequence $$0\mapsto T_{X/Y}\rightarrow T_{X}\rightarrow f^{*}T_{Y}\mapsto 0$$ Now let us assume $f$ to be a ...
Puzzled's user avatar
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3 votes
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Intersections on Blow up

Let $X$ be a smooth variety over $\mathbb{C}$. Blowing up a subvariety $Y\subset X$ of codimension $\ge 2$, we get $\pi: X'\rightarrow X$. Assume $X'$ is smooth and $E$ is the only exceptional divisor....
user40888's user avatar
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1 answer
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Geometric meaning of the positive part of graded ring

Let's say I have a complex projective variety $X\subseteq\mathbb P^n$ with homogeneous coordinate ring $S=\bigoplus_{d\ge 0} S_d$. The localization by some homogeneous $f\in S$ (of nonzero degree) ...
Jesko Hüttenhain's user avatar
12 votes
1 answer
2k views

Why A. Weil considered elimination theory to be eliminated?

It is well known that André Weil declared, in the 1940's, that elimination theory must be eliminated from algebraic geometry. I would like to understand his mathematical reasons to adopt such an ...
Al-Amrani's user avatar
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4 votes
1 answer
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Is blowing down functorial?

Let $f: X \to Y$ be a morphism of smooth surfaces and $C \subset Y$ be a contractible exceptional curve on $Y$. Assume that $f^{-1}(C)$ is a contractible exceptional curve as well. Denote by $X'$ (...
Chen's user avatar
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Complex structure of the Teichmüller space in terms of Fenchel-Nielsen coordinates

The Teichmüller space $T_g$ of genus $g$ Riemann surfaces can be parameterized in terms of Fenchel-Nielsen coordinates, taking values in $\mathbb{R}^{3g-3}\times \mathbb{R}_+^{3g-3}$. The ...
Yuji Tachikawa's user avatar
1 vote
1 answer
239 views

Cotangent bundle

We have the sequence $0 \rightarrow \Omega^1_{\mathbb{P}^2}\rightarrow3\mathcal{O}_{\mathbb{P}^2}(-1)\rightarrow \mathcal{O}_{\mathbb{P}^2}\rightarrow 0$. Can we write a exact sequece such that $\...
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Projective bundles

Fix $n$ and let $0\leftarrow \mathcal{F}\leftarrow \bigoplus \mathcal{O}_{\mathbb{P}^n}(a_i)\leftarrow \bigoplus \mathcal{O}_{\mathbb{P}^n}(b_i)\leftarrow \cdots$ be an exact sequence. Then we can ...
Bajouca's user avatar
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Computing cotangent complex

I would like to know if one can compute all the cohomology sheaves of the cotangent complex of a subvariety of the affine space once a resolution of its ideal sheaf is given? In my precise situation, ...
Libli's user avatar
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7 votes
1 answer
455 views

Spin manifolds with one parallel spinor

Are there any examples of D-dimensional Ricci-flat Riemannian (spin) manifolds of dimension D= 2,3,4,5 with the dimension of the space of parallel spinors equal to 1? And the same question for the ...
Vladimir's user avatar
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5 votes
1 answer
686 views

Cubic forms and Hasse Principle

It's well-known that quadratic forms over the rational numbers $\mathbf{Q}$ satisfy the Hasse-Minkowski theorem. This is to say that they are isotropic over $\mathbf{Q}$ if and only if they are ...
stankewicz's user avatar
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Image of critical points

Let $K$ be a field of characteristic $0$, $f:K^n\rightarrow K^n$ be an algebraic function, that is, $n$ polynomial functions in $n$ variables. Let $S$ be the set of critical points of $f$. If $K=\...
loup blanc's user avatar
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Shimura varieties and Maximal conditions

Working with Shimura varieties, I have been convinced to call them (or the families giving rise to them especially in $A_{g}$) somehow the "maximal" families. The motivation of this, has been for ...
Darius Math's user avatar
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Could we extend the Atiyah class to the sheaf of poly-vector fields to get a Poisson bracket?

Let $X$ be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction of the global existence ...
Zhaoting Wei's user avatar
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18 votes
1 answer
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Geometry behind Rees algebra (deformation to the normal cone)

Let me start with the formal definition of Rees algebra. If $A$ is a commutative ring over some field $k$, $I \subset A$ is an ideal, then Rees algebra is by definition $$ R=\oplus_{i \in \mathbb{Z}} ...
Sasha Pavlov's user avatar
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5 votes
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Differential Geometry of (Non-Abelian) Gerbes in the language of Brylinski

Context In an effort to have a definition for connection on a non-abelian gerbe, in the style of Brylinski, I am reading Breen and Messing's Differential Geometry of Gerbes [BM]. It seems that there ...
cheyne's user avatar
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lefschetz hyperplane theorem in positive characteristic

The proof of the Lefschetz Hyperplane theorem over $\mathbb{C}$ is well-known, and relies on Hodge theory in an important way. Does there exist an analogue of this theorem in positive characteristic, ...
IMeasy's user avatar
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11 votes
2 answers
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Central extension of the algebraic loop group

I'm doing some constructions with the universal central extension $\widehat{\Omega G}$ of the loop group $\Omega G$ (here $G$ is a matrix group), where a priori the loops involved are just smooth, but ...
David Roberts's user avatar
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1 answer
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Can we define the tensor product in the derived category $D^b_{\text{coh}}(X)$ just from $D^b_{\text{coh}}(X)$ in certain cases?

This question arise from the comparision of the reconstruction theorems of Bondal-Orlov and Balmer and is inspired by Shizhuo Zhang's mathoverflow question: How to unify various reconstruction ...
Zhaoting Wei's user avatar
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14 votes
1 answer
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Learning a little Motivic Cohomology

Simply because I find it interesting, I have spent some time studying motivic cohomology from the lectures by Mazza, Voevodsky and Weibel. However, I'm finding it hard to tell if the theory is ...
LMN's user avatar
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674 views

Ample Line Bundles on Algebraic Spaces

The sources known to me (Knutson's Algebraic Spaces and Pascual-Gainza's Ampleness criteria for algebraic spaces) define a line bundle $L$ on an algebraic space $X$ (over a base scheme $S$) to be ...
Lennart Meier's user avatar
2 votes
1 answer
247 views

Height on a semiabelian variety

Let $A$ be a semiabelian variety over $\bar{\mathbb{Q}}$ and $B$ a semiabelian subvariety of $A$. Let $\pi:A\to A/B$ be the canonical morphism. Let $h:A(\bar{\mathbb{Q}})\to\mathbb{R}$ and $h':(A/B)(\...
user40698's user avatar
4 votes
0 answers
252 views

Can one use the equivariant Thom-Gysin sequence on a singular affine variety?

Let $X$ be a smooth complex affine variety. Suppose that $\{X_{\beta}\}_{\beta\in B}$ is a finite stratification of $X$ into smooth locally closed subvarieties. Let $T$ be a complex algebraic torus ...
Peter Crooks's user avatar
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3 votes
1 answer
597 views

Normal form for a holomorphic Morse function

Similarly to Morse lemma, a holomorphic Morse function can be written, near a critical point, as $W_1^2+W_2^2+...+W_n^2+C$ , for well chosen coordinates $W_1,W_2,...,W_n$. I want to cite this result ...
feng's user avatar
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2 answers
275 views

on a characterisation of regular D-modules

Let $X$ be a smooth variety over a field of characteristic zero. Let $M$ be in the derived category of holonomic $\mathcal{D}_{X}$-modules, $D^{b}_{h}(\mathcal{D}_{X})$. We know that if we assume ...
prochet's user avatar
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7 votes
1 answer
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Brauer groups of punctured affine lines over a base

Let $R$ be a torsion-free regular noetherian ring. The Brauer group $Br(R)$ of $R$, defined equivalently (by a theorem of Gabber) as the group of Morita equivalence classes of Azumaya $R$-algebras or ...
Akhil Mathew's user avatar
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4 votes
1 answer
479 views

Weighted projective space with rational or real weights

The most common formulation of the weighted projective space is perhaps the global quotient $$ (\mathbb{C}^{n+1} \setminus \{(0,\ldots,0\}) / \mathbb{C}^\ast $$ with the $\mathbb{C}^\ast$ group action ...
ssquidd's user avatar
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4 votes
2 answers
1k views

Effective divisors on the blow-up of $\mathbb{P}^n$

Let $X^n_r$ the blow-up of $\mathbb{P}^n$ at $r$ points in very general position. (1) It is known in general the Effective Cone or the Numerical Effective Cone of those algebraic variety? Let $X$ ...
Joaquín Moraga's user avatar
16 votes
1 answer
1k views

When is $f(x_1, \dots, x_n)+c$ an irreducible polynomial for almost all constants $c$?

Let $f\in \mathbb C[x_1, \dots, x_n]$, $n\ge 1$, be a non-constant polynomial. Consider the polynomial $f+t\in \mathbb C[t, x_1,\dots, x_n]$. This is an irreducible polynomial in $\mathbb C(t)[x_1, \...
Cantlog's user avatar
  • 550
4 votes
0 answers
464 views

Points of moduli space of semistable sheaves and S-equivalence classes

Let $X$ be smooth projective and connected curve over an algebraically closed field $k$. One knows the description of $k$-valued points of the moduli space $M_X$ of semistable vector bundles of fixed ...
boxdot's user avatar
  • 203
4 votes
1 answer
391 views

degrees of complex projective spaces and quadrics

A well-known result of Kobayashi and Ochiai says that an $n$-dimensional Fano maniofold $M$ is biholomorphic to $\mathbb{C}P^n$ or complex quadrics if its index is $n+1$ or $n$ respectively. In these ...
Kevin's user avatar
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7 votes
0 answers
649 views

Etale cohomology of Berkovich spaces

Suppose $X/\mathbb{Q}$ is a reasonable smooth projective variety with interesting etale cohomology. For example, we can say $X$ is an elliptic curve. To what extent does it make sense to study the ...
none's user avatar
  • 71
1 vote
1 answer
599 views

What was the original/historical motivation for introducing Grothendieck (pre-)topologies

The title essentially explains it, but I'll give some background: I'm giving a talk to some fellow grad students about the relative Picard functor which requires introducing Grothendieck (pre-)...
John Stalfos's user avatar
2 votes
1 answer
262 views

global sections of higher direct images of étale sheaves

Is there a useful criterion for when $\Gamma(X, R^qf_*F) = H^q(X',F)$, $f: X' \to X$, $F$ an étale sheaf on $X'$?
user avatar
8 votes
1 answer
378 views

Action of the endomorphism monoid on an irreducible GL-module

Let $G=\mathrm{Gl}_n(\mathbb C)$ and $V$ an irreducible $G$-module on which $G$ acts polynomially. In other words, the algebraic group action of $G$ on the affine space $V$ extends to an algebraic ...
Jesko Hüttenhain's user avatar
9 votes
0 answers
517 views

Getting a bound via polynomial equations

When studying the existence problem of power residue deference sets, I came across the following system of polynomial equations over $\mathbb{C}$, \begin{cases} &\sum\limits_{j=0}^{m-1}x_jx_{2k-j}=...
Binzhou Xia's user avatar
1 vote
0 answers
396 views

Weakened jacobian conjecture for entire functions

A rudin's theorem is the assertion that any polynomial injection between affine spaces of the same dimension has a polynomial inverse, and the inverse is also given by polynomials. The jacobian ...
Koushik's user avatar
  • 2,076
15 votes
4 answers
2k views

What is the relation between the Lie bracket on $TX$ as commutator and that coming from the Atiyah class?

Let X be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction to the global existence ...
Zhaoting Wei's user avatar
  • 8,657
2 votes
1 answer
1k views

relation between sheaf of hom and hom of sheaf

If $\mathcal{M,N}$ are the associated sheaf of $A$ modules $M$ and $N$ on $X=Spec A,$ then what is $\mathcal{Hom_{O_X}(M,N)}$?Is that the associated sheaf of $Hom(M,N)\ ?$
user40534's user avatar
10 votes
1 answer
2k views

Equivariant resolution of singularities

I am looking for some references on equivariant resolution of singularities. In most references quoted on mathoverflow (for instance : Reference on an equivariant resolution of singularities), they ...
Libli's user avatar
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2 votes
0 answers
318 views

PAC field : Algebraically closed field :: ? : Henselian local ring

I'm wondering if the following exists in the world as a definition. I'll use the word "pseudo-Henselian." I'll restrict to DVRs for simplicity. I'd want to call a DVR $(R,\mathfrak{m})$ pseudo-...
Bobby Grizzard's user avatar
7 votes
3 answers
719 views

Does every linear group admit a subgroup of dimension 1?

Suppose that $G$ is a linear group of positive dimension, defined over some field $k$. Is that true, that $G$ admits a (closed) one-dimensional subgroup? I'm pretty much sure this is true in ...
Tomasz Lenarcik's user avatar
3 votes
1 answer
380 views

Existence of quotient variety for group implies existence of quotient for normal subgroups

Let $G=G_1\times G_2$ be a product of two linear algebraic groups over an algebraically closed field. Assume that $G$ acts on a variety $X$ such that the quotient $X\rightarrow X/G$ exists in the ...
Jan Weidner's user avatar
  • 12.8k
3 votes
2 answers
395 views

Quotient of an abelian surface by an antisymplectic involution

What can we say about the quotient of an abelian surface by an antisymplectic involution?
sqrt2sqrt2's user avatar
0 votes
1 answer
206 views

Totally tangent planes to a curve in $\mathbb{P}^3$

How many planes are totally tangent to a curve of $\mathbb{P}^3$ which is intersection of a generic quadric and a generic cubic? Or equivalently, considering the cubic as a blow up of $\mathbb{P}^2$ ...
sqrt2sqrt2's user avatar
6 votes
2 answers
720 views

The fundamental group of a complex, quasi-affine variety

Can the fundamental group of a quasi-affine variety over $\mathbb{C}$ be a torsion group?
IMeasy's user avatar
  • 3,717
4 votes
1 answer
418 views

word problem for the fundamental group of complements

It is well known that the finite type (pure) Artin groups have solvable word problem. This was proved by Deligne in 1972. His aim was to show that the complement of a simplicial hyperplane arrangement ...
Priyavrat Deshpande's user avatar
7 votes
2 answers
348 views

Do taking a general hyperplane section and taking a colon ideal commute?

Let $I$ be an ideal and $f$ be an element of $R = \mathbb{C}[x_1,\ldots,x_n]$, where $\mathbb{C}$ is an algebraically closed field of characteristic $0$. Does $$ (I+H):f = (I:f)+H $$ hold for a ...
Anton Leykin's user avatar

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