Questions tagged [ag.algebraic-geometry]
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
21,545
questions
3
votes
2
answers
600
views
Real analytic submanifolds of $\mathbb{R}^{n}$
Hallo,
Let $(M,g)$ be a Riemannian $k$-dim real analytic submanifold of $\mathbb{R}^{n}$. Is it true that $M$ in $\mathbb{R}^{n}$ looks locally (in a small neigbourhood around some point in $M$) as ...
15
votes
0
answers
586
views
For how many primes does an elliptic curve over a totally imaginary field have supersingular reduction?
An elliptic curve over a finite field, $k$, of characteristic p is called supersingular if it has no $p$-torsion over $k^{\mathrm{alg}}$, or equivalently, if $\mathrm{End}(E)$ is an order in a ...
3
votes
2
answers
709
views
Higgs bundle and stable bundle
Let $(E,\phi)$ be a $G$-Higgs bundle $\phi\in H^{0}(X,ad(E)\otimes D)$ where $D$ is a divisor on X.
I suppose that $(E,\phi)\in \mathcal{M}^{ani}$ the anisotropic locus.
In particuler, this bundle ...
36
votes
3
answers
6k
views
Conjectures in Grothendieck's "Pursuing stacks"
I read on the nLab that in "Pursuing stacks" Grothendieck made several interesting conjectures, some of which have been proved since then. For example, as David Roberts wrote in answer to ...
2
votes
0
answers
469
views
$\sigma$-conjugate iff $p$-adically close
First some notations. Let $p$ be a prime, $k$ a perfect field of characteristic $p$, $W=W(k)$ the ring of Witt vectors over $k$, $\sigma : W \rightarrow W$ the Frobenius, $R$ a commutative $\mathbb{Z}...
0
votes
0
answers
801
views
restriction and pullback of representable etale sheaf along closed immersion
I find that the restriction and pullback of representable etale sheaf along closed immersion are very confusing. I think they are different in general, I hope some experts can confirm my understanding ...
6
votes
0
answers
476
views
Why is algebraic de Rham cohomology via completion independent of embedding?
In Hartshorne's "On the de Rham cohomology of algebraic varieties", he defines algebraic de Rham cohomology of a variety $X$ over a field $k$ of characteristic zero by choosing a closed immersion $X \...
12
votes
1
answer
1k
views
On the derived category of constructible étale sheaves
The derived category $D^{\flat}_{c}(X,R)$ of constructible sheaves of $R$-modules on $X_{et}$ is defined as the full subcategory of $D^b(X,R)$ whose cohomology sheaves are all constructible.
Clearly, ...
8
votes
4
answers
986
views
Subspaces of End(V) that can fix any vector
Suppose V is a finite-dimensional vector space and I have a linear subspace of its endomorphisms
$$W \subseteq \mbox{End}(V).$$
How can I easily check if every vector of $V$ is fixed by some element ...
2
votes
2
answers
1k
views
Equation for simple Jacobian of a genus two curve
Let $X$ be a curve of genus two over a field $k$ with a $k$-rational point. Let $J$ be the Jacobian of $X$.
Can we write down an explicit equation for the abelian surface $J$?
I know $X$ can be ...
58
votes
3
answers
4k
views
What is the geometry of an undecidable diophantine equation?
As an arithmetic algebraic geometer of the highest moral fiber, I am trained to look at Diophantine equations in terms of the geometry of the corresponding scheme. For instance, if the Diophantine ...
3
votes
1
answer
202
views
Detecting sections on an arithmetic variety
Let $S$ be Spec $O_K$ with $O_K$ the ring of integers of a number field $K$.
Let $X\to S $ be an arithmetic variety, i.e., an integral smooth quasi-projective $S$-scheme with generic fibre $X_\eta$ ...
8
votes
1
answer
851
views
When is the kernel of the etale fundamental group in a fibration abelian?
Let $X \to Y$ be a smooth proper morphism. Let $y$ be a geometric point of $Y$. Is the kernel of the natural map of etale fundamental groups $\pi_1^{et}(X_y) \to \pi_1^{et} (X)$ abelian?
This is true ...
1
vote
1
answer
1k
views
cohomology of torsion sheaves and nilpotent sheaves
Let $X$ be a scheme and $\mathcal{F}$ be a sheaf on $X$ which is torsion $\mathcal{O}_X-$module (i.e., every local section is annihilated by an element of the ring $\mathcal{O}_X(U)$) or nilpotent (i....
3
votes
0
answers
195
views
Jacobians for arithmetic curves
We know that the Jacobian of an algebraic curve play an important role in the study of curves. My question is: Is there a "Jacobian" for an arithmetic curve such as $Spec Z$, which parameterizes some ...
14
votes
0
answers
2k
views
conformal blocks for beginners
I have given now a couple of talks that involve conformal blocks bundles on the moduli stack $\overline{\mathcal{M}}_{g,n}$, in front of a public of algebraic geometers but not specialists of the ...
2
votes
3
answers
485
views
Minimum 1st-neghbors distance between N random points on a ring
We have $N$ points randomly and uniformly distributed on a ring of length 1.
Let $d_i$ be the distance between point $i$ and its first neighbor.
We want to know the expected value of the smallest $...
2
votes
1
answer
349
views
Techniques for showing that a curve is not smoothable
There are a number of techniques in algebraic geometry that can be used to show that a given reducible (often genus-zero) curve $C$ in a smooth variety $X$ becomes smooth and irreducible after a (...
1
vote
2
answers
191
views
Can a classifying space be characterised universally? [closed]
I'm having trouble understanding what classifying spaces are in general.
It seems to me, that they are terminal in a category of bundles whose morphisms are pullback squares, is this correct?
5
votes
1
answer
800
views
Properties of subvarieties of a simple abelian variety
Let $A$ be a simple abelian variety over a field $k$. (For simplicity, we assume char $k =0$.)
Let $X$ be a smooth projective geometrically connected variety over $k$ of positive dimension.
Suppose ...
25
votes
2
answers
2k
views
Profinite groups as étale fundamental groups
Does every profinite group arise as the étale fundamental group of a connected scheme?
Equivalently, does every Galois category arise as the category of finite étale covers of a connected scheme?
...
4
votes
1
answer
166
views
is there some condition I can impose on families of curves on a surface such that the second Ext between the ideal sheaf and the structure sheaf is zero?
(the title got out of hand)
Say I have a surface $X$, then I also have M, the Hilbert scheme of curves and points on X.
This can be seen as a moduli space of quotients $O_X \to O_Z$.
If $I_Z$ is the ...
2
votes
2
answers
282
views
Relating the toric rank of a semistable curve and the first Betti number of its reduction graph
Introduction
Let $k$ be a local field. Let $C$ be the spectrum of $\mathcal{O}_{k}$. Let $X/k$ be a smooth projective curve with a semistable model $\mathcal{X}/C$.
Let $J$ be the Jacobian of $X$. ...
2
votes
0
answers
557
views
tangent bundle of the toric variety of the wonderful compactification.
Let G be a adjoint group over $k$,algebraically closed of caracteristic zero.
Let $\overline{G}$ be its wonderful compactification.
I denote by $\overline{T}$ the closure of the torus $T$ in $\...
32
votes
4
answers
4k
views
Open problems in Birational Geometry, after BCHM
Rencently a breakthrough was made in the context of the Minimal Model Program by the work of Birkar-Cascini-Hacon-McKernan. They proved that the canonical ring of a smooth or mildly singular ...
4
votes
1
answer
969
views
Equations of the Hirzebruch surface embedded in a large space.
Let $\mathbb{F}_1$ be the Hirzebruch surface $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(-1))$ and let $D$ be the very ample divisor $3C_0+5f$ on $\mathbb{F}_1$ (notation as in [Hartshorne, Algebraic ...
8
votes
1
answer
1k
views
Tensor product of $\mathcal{D}$-modules and constructible sheaves
The Riemann-Hilbert correspondence, as proved by Kashiwara and Mebkhout, says that for X a smooth algebraic variety over $\mathbb{C}$ there is an equivalence of triangulated categories
$D^b_c(X,\...
5
votes
3
answers
923
views
algebraic groups and their Lie algebras
I have probably a stupid question about representations of algebraic groups:
Let $G$ be an algebraic group and $L$ be a Lie algebra of $G$. What is the connection between
categories of ...
10
votes
2
answers
2k
views
Tangent space of the stack $\overline{\mathcal{M}}_{g,n}(X,\beta)$.
Let $\overline{\mathcal{M}}_{g,n}(X,\beta)$ be the moduli stack of stable maps $f$ from genus $g$, $n$-marked curve $C$ to a variety $X$ i nth curve class $\beta \in H^2(X,\mathbb{Z})$. I would like ...
3
votes
1
answer
333
views
plane cubics and conic bundles
It is well known that any plane cubic curve can be obtained as the discriminant locus of a conic bundle (actually even just of a net of conics). Does this hold true also for all nodal cubics (with ...
0
votes
1
answer
186
views
Elementary transformations and determinant maps.
Let $S$ be a smooth projective surface and $C$ a smooth, irreducible curve contained in $S$. Let $E_1$ and $E_2$ be two vector bundles on $S$ having the same rank and assume they lie in a short exact ...
7
votes
2
answers
2k
views
Number of irreducible and connected components constant in flat families
A) Let $f:F\rightarrow S$ be a flat proper morphism of schemes with geometrically normal fibers. Then supposedly the number of $\textbf{connected}$ components of the geometric fibers is constant. ...
4
votes
1
answer
449
views
Disconnectedness of Hilbert schemes of projective schemes
Let $Y$ be a projective scheme. The naive definition of a Hilbert scheme of subschemes $X$ of $Y$ would require us to projectively embed $Y$, then ask that $X$ have a fixed Hilbert polynomial $p$.
...
6
votes
0
answers
175
views
Perturbations of zero-dimensional algebraic varieties
Let $P(z,w):\mathbb C^2\to\mathbb C$ be a certain polynomial, and consider $p(s,t)=P(e^{is},e^{it}):\mathbb T^2\to \mathbb C$ its restriction to the real torus. Assume generic situation, so by ...
3
votes
0
answers
643
views
An example of almost etale extension
In the paper of Faltings' "p-adic Hodge theory", Faltings showed an example of almost etale extension before he proved the almost purity theorem. The example is following:
Let $k$ be a perfect field ...
8
votes
2
answers
2k
views
Equations of the secant variety
Let $X\subset\mathbb{P}^N$ be an irreducible nondegenerate (i.e. not contained in a hyperplane) projective complex algebraic variety, and let $\mathrm{Sec}(X)$ be the secant variety of $X$ (i.e. the ...
9
votes
0
answers
847
views
Fontaine-Mazur for Hodge Structures
Is there a conjecture, or known result, describing which integral Hodge structures are composition factors in the Hodge structure on the cohomology groups of smooth proper algebraic varieties over $\...
6
votes
1
answer
2k
views
Scheme defined over $\mathbb{Z}$
I'd like to check a definition:
If $X$ is a scheme, what does it mean to say that $X$ is "defined over $\textrm{Spec }\mathbb{Z}$"? Is this a precise statement? Certainly this statement ...
2
votes
1
answer
154
views
Need references on moduli of subvarieties of given dimension and degree of the projective space P^N
Hello,
I need to understand the moduli of subvarieties of given dimension and degree of the projective space P^N. Is there any good place to look into? Thank you.
4
votes
1
answer
875
views
Abelianized fundamental group of a curve over a finite field
Let $X$ be a smooth, projective, and geometrically connected curve over a finite field $\mathbb{F}_q$ and fix a geometric point $\overline{x} : \text{Spec } \overline{\mathbb{F}_q} \to X$. Then there ...
2
votes
0
answers
681
views
On relative dualizing sheaf
Let $f: X \to C$ be a fibration from a smooth variety $X$ to a smooth curve $C$ over an algebraically closed field $k$. If $k=\mathbb C$, we know for all $i$, $R^i f_* \omega_{X/C}$
is semipositive, ...
1
vote
1
answer
546
views
finite surjective morphism to the projective line
Let X a smooth projective curve over $\mathbb{C}$.
We fix $d$ distinct closed points $x_{1},\dots,x_{d}$.
Can we find a finite surjective morphism $\pi:X\rightarrow\mathbb{P}^{1}$
and local ...
0
votes
1
answer
117
views
Reference request: base point freeness of $2\Theta$
Let $J$ be a Jacobian variety defined over a field $k$ and let $\Theta$ be a symmetric theta-divisor on $J$.
It's shown (for instance) in the book Complex Abelian Varieties by Lange and Birkenhake ...
41
votes
1
answer
18k
views
What is "Teichmüller Theory" and its history?
What is "Teichmüller Theory"? What part has been worked out / foreseen by O. Teichmüller himself and what is further development? Is there some current work which might be considered as continuation/...
3
votes
0
answers
118
views
Extending cohomology classes to compactifications of Kuga varieties
I am trying to understand the proof of lemma 3 in the paper "Algebraic cycles and the Hodge structure of a Kuga fiber variety" by B. Brent Gordon,
available at http://www.ams.org/journals/tran/1993-...
0
votes
0
answers
396
views
Normal sheaf of non-reduced locally complete intersection space curves
Let $C$ be a non-reduced locally complete intersection curve on a smooth degree $d$ surface in $\mathbb{P}^3$ (for example a non-reduced Cartier divisor). For simplicity we can assume that $d>deg(C)...
13
votes
1
answer
2k
views
What is the universal deformation of the formal additive group $\widehat{\mathbb{G}}_a$ over $\mathbb{F}_p$?
Lubin and Tate show in their paper Formal moduli for one-parameter formal Lie groups that for any formal group over a field $k$ of characteristic $p>0$ with height $h<\infty$, the functor of ...
7
votes
2
answers
653
views
$n$-path-connected components of a variety
This question is motivated by the question Path Connectedness of Varieties and some funny little theorem I was trying to prove. Let $X$ be a (quasiprojective smooth connected) algebraic variety over ...
2
votes
0
answers
215
views
ALE Kähler manifolds are birational to deformations of $\mathbb{C}^n/G$.
I am reading Dominic Joyce's book 'Compact Manifolds with Special Holonomy' and I am struggling to understand a remark he makes at the end of chapter 8. The assertion is (I think) the following:
...
7
votes
2
answers
508
views
restriction of the cotangent bundle of an elliptic surface
Given an nonisotrivial elliptic fibration $f:X\rightarrow P^1$, where $X$ is smooth and $P^1$ is a projective line. Could anybody provide some information on the restriction of the cotangent bundle $\...