Questions tagged [ag.algebraic-geometry]

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

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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 ...
hapchiu's user avatar
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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 ...
Lloyd Yu-West's user avatar
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 ...
prochet's user avatar
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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 ...
AAK's user avatar
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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}...
user26756's user avatar
  • 271
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 ...
Heer's user avatar
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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 \...
A. Pascal's user avatar
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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, ...
David Corwin's user avatar
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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 ...
John Wiltshire-Gordon's user avatar
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 ...
Mike Lowrey's user avatar
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 ...
Will Sawin's user avatar
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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 $X_\eta$ ...
Mike Lowrey's user avatar
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 ...
Will Sawin's user avatar
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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....
Naga Venkata's user avatar
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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 ...
Diego Maradona's user avatar
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 ...
IMeasy's user avatar
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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 $...
Luce's user avatar
  • 23
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 (...
Charles Staats's user avatar
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?
Mozibur Ullah's user avatar
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 ...
Mike Lowrey's user avatar
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? ...
Martin Brandenburg's user avatar
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 ...
Jacob Bell's user avatar
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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 $X$. ...
jmc's user avatar
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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 $\...
prochet's user avatar
  • 3,432
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 ...
gio's user avatar
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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,\...
ChrisLazda's user avatar
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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 ...
Andriy Regeta's user avatar
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 ...
Zheng's user avatar
  • 101
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 ...
bugger's user avatar
  • 33
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 ...
ginevra86's user avatar
  • 763
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. ...
HNuer's user avatar
  • 2,098
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 $p$. ...
Allen Knutson's user avatar
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 ...
dima's user avatar
  • 949
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 ...
kiseki's user avatar
  • 1,911
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 ...
gio's user avatar
  • 1,149
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 $\...
Will Sawin's user avatar
  • 135k
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 ...
LMN's user avatar
  • 3,525
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.
nono's user avatar
  • 165
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 ...
Justin Campbell's user avatar
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, ...
Tong's user avatar
  • 575
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 ...
prochet's user avatar
  • 3,432
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 ...
jsm's user avatar
  • 337
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/...
Alexander Chervov's user avatar
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-...
A Confused Cat's user avatar
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)...
Naga Venkata's user avatar
  • 1,010
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 ...
Aaron Mazel-Gee's user avatar
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 ...
Bugs Bunny's user avatar
  • 12.1k
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: ...
oydeis's user avatar
  • 21
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 $\...
gummi's user avatar
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