Questions tagged [ag.algebraic-geometry]
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
21,545
questions
3
votes
0
answers
245
views
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_{...
2
votes
2
answers
727
views
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?
0
votes
1
answer
309
views
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 ...
3
votes
0
answers
185
views
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....
4
votes
1
answer
550
views
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) ...
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 ...
4
votes
1
answer
524
views
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 $X'$ (...
13
votes
3
answers
1k
views
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 ...
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 $\...
0
votes
1
answer
234
views
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 ...
7
votes
1
answer
2k
views
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, ...
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 ...
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 ...
0
votes
0
answers
255
views
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=\...
4
votes
0
answers
190
views
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 ...
3
votes
1
answer
366
views
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 ...
18
votes
1
answer
3k
views
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}} ...
5
votes
0
answers
398
views
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 ...
1
vote
0
answers
572
views
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, ...
11
votes
2
answers
2k
views
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 ...
7
votes
1
answer
1k
views
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 ...
14
votes
1
answer
2k
views
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 ...
4
votes
0
answers
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 ...
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)(\...
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 ...
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 ...
3
votes
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 ...
7
votes
1
answer
1k
views
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 ...
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 ...
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$ ...
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, \...
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 ...
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 ...
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 ...
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-)...
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'$?
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 ...
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}=...
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 ...
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 ...
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)\ ?$
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 ...
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-...
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 ...
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 ...
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?
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$ ...
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?
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 ...
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 ...