Questions tagged [ag.algebraic-geometry]

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

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Do $PGL_n$-torsors induce elements of the Brauer group

Let $K$ be a field and let $n\geq 2$. If $n=2$, then the set of $K$-isomorphism classes of $PGL_n$-torsors is in bijection with the $n$-torsion of the Brauer group of $K$. Is this only for $n=2$? Is ...
brauer's user avatar
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7 votes
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Reference result: proof of theorem of Kazhdan-Margulis on monodromy group of a Lefschetz pencil of odd fiber dimenion is "as big as possible"

In Deligne's paper on his first proof of the Weil conjectures, we have the following result. Theorem 5.10 (Kazhdan-Margulis). L'image de $\rho: \pi_1(U, u) \to \text{Sp}(E/(E \cap E^\perp), \psi)$ ...
user avatar
5 votes
1 answer
247 views

Degree of irrationality and hyperelliptic curves

For a variety $V$ of dimension $n$, let $Irr(V)$ denote the minimal degree of a dominant rational map $V\to \mathbb{P}^n$. Suppose that a curve $X$ admits a dominant map from a variety $V$ with $...
Nico Bellic's user avatar
6 votes
1 answer
1k views

Smoothness of the branch divisor and ramification on surfaces

Let $f \colon X \longrightarrow Y$ be a finite morphism of degree $n \geq 2$ between smooth compact, complex surfaces. Let $B \subset Y$ be the branch divisor of $f$ and assume that the ...
Francesco Polizzi's user avatar
1 vote
1 answer
212 views

coarse moduli space $X(2)$

Let $\mathfrak{M}(2)$ be the algebraic stack over $\mathbb{Z}[1/2]$ which classifies the elliptic curves with the $\Gamma(2)$ level structure and let $M(2)$ be its coarse moduli space. Is there an ...
Adel BETINA's user avatar
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10 votes
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Algebraic geometry and PDEs (reference-request)

Context: Let's say we have an affine algebraic variety corresponding to the zero set of an irreducible polynomial (over $\mathbb{C}$) in $n$ variables, denoted by $p(x_1, \dots, x_n)$. $$p(x_1, \dots, ...
Chill2Macht's user avatar
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2 votes
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Has this notion of powers of ideals already appeared in the literature?

My question is whether the below notion of powers of ideals, which may be regarded as a weakening of the notion of symbolic power, has been already defined in the literature and, in that case, I ask ...
Alberto Fernandez Boix's user avatar
1 vote
1 answer
104 views

Actions of torsionfree discrete subgroups on hermitian symmetric domains

Let $D$ be a bounded hermitian symmetric domain with automorphism group $G(\mathbb R)$. In the example I have in mind, $D$ is Siegel upper half-space of degree $g$ and $G(\mathbb R) = \mathrm{Sp}(2g,\...
John's user avatar
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1 vote
1 answer
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Discontinuous subgroups of $PGL_2(\mathbb{Q}_p)$

I'm trying to read about Mumford curves. I've barely begun and I've already encountered a stumbling block. I'm sure this is probably a basic question that an expert could resolve quickly. I would very ...
stupid_question_bot's user avatar
6 votes
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If $X$ is a genus $g\geq 2$ curve over a number field $K$, then is there a bound on $X_L(L)$ for $L/K$ that depends only on $X$ and $[L:K]$?

If $X$ is a genus $g\geq 2$ curve over a number field $K$, then $X(K)$ is finite by Falting's Theorem. My question is how does $X_L(L)$ behave for finite field extensions $L/K$? In particular, is ...
Andrew NC's user avatar
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1 answer
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Relationship between motivic Galois groups and Langlands program [duplicate]

I would like to know if there is any relationship between the motivic Galois groups and the Langlands program. Many thanks.
tttbase's user avatar
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5 votes
1 answer
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Supposed generalization of $X/(G \times H)\simeq (X/G)/H$ for GIT-quotients

I wonder whether it is true that the composition of two GIT-quotients is another GIT-quotient. It should be an analogue of a set-theoretic formula $X/(G \times H)\simeq (X/G)/H$ but with GIT-quotients ...
evgeny's user avatar
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4 votes
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An example of a commutative ring with infinitely many maximal ideals

I am looking for a commutative ring $R$ with identity that has the following properties: 1) $\mathrm{Max}(R)$, the set of all maximal ideals of $R$, is infinite; 2) whenever $\{I_\alpha\}_{\alpha\in\...
Andro Zimone's user avatar
3 votes
1 answer
151 views

Taking powers of polytopes

I am not sure this is a well framed question but I would like to know if anything like "taking the power" of a polytope is known. Imagine this situation where I want to think of such a thing : say ...
gradstudent's user avatar
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4 votes
1 answer
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Proof of Theorem 6.8 in the paper "Singular homology of abstract algebraic varieties"

Background: Theorem 6.8 in Suslin and Voevodsky's article "Singular homology of abstract algebraic varieties" states that there is an isomorphism between effective relative zero cycles $z_0^c(Z)^{eff}(...
Oliver E. Anderson's user avatar
0 votes
2 answers
391 views

Group actions on affine space which are almost good

Let $G$ be a finite group acting on $\mathbb A^n_{\mathbb C}$. Let $Y$ be a dense open whose complement is of codimension at least two. Assume $Y$ is $G$-stable, the action of $G$ is free on $Y$, ...
Miele's user avatar
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5 votes
5 answers
648 views

Relation between $h^1(\mathcal{O}_X)$, $h^0(\Omega_X)$, and first betti number for general complex manifold?

I was wondering in the world of non-Kähler compact complex manifold, what is the relationship between $h^1(\mathcal{O}_X)$, $h^0(\Omega_X)$, and first betti number? The only thing I know is that for ...
S. Li's user avatar
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2 votes
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If a subgroup H of a finite group G acts freely on a variety, can the G-Hilbert scheme be computed by iteration?

Let $X$ be a smooth quasi projective variety over $\mathbb{C}$. Let $G$ be a finite abelian group acting via automorphisms on $X$. Denote by $G$-$\text{Hilb}(X)$ the subscheme of the Hilbert scheme ...
Bernie's user avatar
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3 votes
1 answer
271 views

K-groups of strict henselization of stalks

How well are the algebraic K-groups of the strict henselization of the stalks $\mathcal{O}_{X,p}^{sh}$ at geometric points of a scheme $X$ understood? I am particularly interested in the case of ...
user's user avatar
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1 vote
0 answers
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Base locus of the Eigen spaces of global sections of totally symmetric line bundle

Let $X$ be an abelian surface over complex numbers. Let $L$ be a totally symmetric line bundle of type $(r,r)$ for $r\geq 2$. The involution $i$ on $X$ gives an action on $H=H^0(X,L)$ thus giving a ...
user52991's user avatar
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1 answer
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Example of finite order symplectomorphism which is not an automorphism

Let $X$ be a smooth projective variety over $\mathbb{C}$. Consider the Kahler structure $(J,g,\omega)$ on $X$ induced by the Fubini-Study metric. Let $Symp(X,\omega)$ be the group of ...
Nick L's user avatar
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1 vote
0 answers
217 views

General approach: to prove finiteness of cohomology theory

This is a rather general (and vague) question. In all cohomology theory, the finitenss result is one of the central things to prove. My question is this: Is there any pattern (unifying structure) ...
guest1234's user avatar
2 votes
0 answers
185 views

How can I find a basis of $H^0(Y, -K_Y)$ for the del Pezzo surface $Y$ of degree 5?

Consider the surface $X \subset \mathbb{P}^2_{x, y, z}\times\mathbb{P}^1_{t, s}$: $$ s^3y^2 + t^3yz = (t+s)x^2 + tsxz + t(t^2 +s^2)z^2 $$ over a finite field $k = \mathbb{F}_{2^d}$, $\mathrm{gcd}(d, 6)...
Dimitri Koshelev's user avatar
3 votes
0 answers
131 views

Arithmetic version of "Attaching maps" for moduli of curves

I am looking for a reference for attaching maps of moduli of curves with marked points. Especially I would like to know whether they descend over $\mathbb{Z}$. On one hand this seems very hard to ...
Bear's user avatar
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3 votes
1 answer
540 views

Dimension of the zero weight space in $V_{2\rho}$

Let $\rho$ be the half sum of the positive roots (also the sum of fundmental weights) for $SL_n(\mathbb C)$. Then $2ρ$ is in the root lattice. Then what is the dimension of the zero weight space for ...
Jack's user avatar
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1 vote
1 answer
151 views

Nature of morphism between Brill Noether varieties

Let $C$ be a smooth curve over complex numbers. Consider the Brill Noether varieties. If $g$ is the genus of $C$. If $r,d$ are positive integers, $$W^r_d=\{A\in Pic^d(C): h^0(A)\geq r+1\},$$ $$G^r_d=\...
user52991's user avatar
  • 159
2 votes
0 answers
159 views

Stable reduction and properness of moduli stacks

I'm reading the notes: https://arxiv.org/pdf/1207.1048.pdf on stable reduction, where on page 31, it's asserted that the Stable Reduction Theorem (Theorem 6.1 on page 30), is equivalent to the ...
stupid_question_bot's user avatar
0 votes
1 answer
127 views

Radical of modules [closed]

Let $R$ be a local ring with the unique maximal ideal ${\frak m}_R$ and $M$ be a $R$-module. Define $I(M) \colon= \cap ~({\mathrm{all~ proper~ maximal ~submodules~ of}}~M)$, where proper means ...
Pierre MATSUMI's user avatar
8 votes
1 answer
606 views

Higgs bundles and stable vector bundle

Let $\mathcal M_X(r,0,K_X)$ be tha moduli space of semistable Higgs bundles over a smooth irreducible algebraic curve over $\mathbb C$. And let $E$ be a stable vector bundle of rank $r$ and degree $0$....
Z.A.Z.Z's user avatar
  • 1,871
4 votes
1 answer
381 views

Representability of relative Hilbert and Picard functors over analytic spaces

Let $f:X \to S$ be a morphism of complex analytic spaces. Then, just like in the case of schemes, we can define the relative Hilbert and Picard functors. For instance, if $\text{An}_{/S}$ denotes de ...
user avatar
3 votes
0 answers
617 views

Roadmap to "Lego Teichmuller"

From a basic understanding of algebraic geometry (on the level of Gathmann's old lecture notes), algebraic number theory (on the level of the first chapter of Neukirch's Algebraic Number Theory), and ...
Anton Hilado's user avatar
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0 votes
0 answers
118 views

determinantal hypersurface singular in codimension 1

I know that the general linear symmetric determinantal hypersurface is singular in codimension 2. Questions. (1) What can we say about linear symmetric determinantal hypersurfaces singular in ...
user46071's user avatar
  • 325
2 votes
1 answer
138 views

Nonnormal locus of cubic hypersurface

If $X\subset \mathbb{P}^{n}$ is a cubic hypersurface that is not normal, what's the easiest way to see that the nonnormal locus is a linear subspace of dimension $n-2$? As for a reference, there is ...
DCT's user avatar
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10 votes
1 answer
473 views

Properties of the petit Zariski topos

What are some (intrinsically formulated) properties of the locally ringed topos $(\mathbf{Sh}(X),\mathcal{O}_X)$ for some scheme $X$, which do not hold for arbitrary locally ringed toposes? Is there, ...
HeinrichD's user avatar
  • 5,402
9 votes
1 answer
836 views

What are the $j$-invariants of the genus 1 modular curves?

I believe there are only finitely many congruence subgroups $\Gamma\le SL_2(\mathbb{Z})$ such that the compactification of $\mathcal{H}/\Gamma$ is genus 1. Is there somewhere I can find a list of ...
stupid_question_bot's user avatar
2 votes
0 answers
487 views

Comparison theorem between étale and de Rham cohomology for local system

This question is based on Milne "canonical models of Shimura varieties and automorphic vector bundles" Let $(G,X)$ be a Shimura datum, $(V,\xi)$ be a rational representation of $G$ (I guess it means ...
user99616's user avatar
1 vote
1 answer
234 views

Analogy of a Fano manifold with anticanonical divisor

Some people say that a Fano manifold with anticanonical divisor is an analogue of a manifold with boundary. Where does this intuition come from?
Noah's user avatar
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4 votes
1 answer
819 views

Hirzebruch-Riemann-Roch theorem for Riemann surfaces with boundary

I would like to know if the Hirzebruch-Riemann-Roch theorem exists for bundles over Riemann surfaces with a boundary. I am asking this because the Hirzebruch-Riemann-Roch theorem is used in the ...
Mtheorist's user avatar
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6 votes
1 answer
2k views

Algebraic vs. homological equivalence for curves on a smooth complex projective surface

Let $X$ be a smooth projective surface over $\mathbb{C}$. Then there is the exponential sheaf sequence: $$ 0 \rightarrow \mathbb{Z} \rightarrow \mathscr{O}_X \rightarrow \mathscr{O}_X^\times \...
dorebell's user avatar
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10 votes
1 answer
496 views

Does GAGA hold over other topological fields?

If k is a non-discrete topological field, we can define an analytic space over k just like complex analytic spaces over $\mathbb{C}$. If you replace "complex analytic space" and "complex algebraic ...
Alex Mennen's user avatar
  • 2,090
1 vote
0 answers
196 views

(Ordered) Configuration space in algebraic geometry

Let $X$ be a topological space and denote by $F_n(X)$ the following subspace: $$F_n(X):=\{(x_1,\cdots ,x_n)\in X^n: x_i\neq x_j \forall i\neq j\}.$$ Note that, we are not considering the quotient of $...
I.P's user avatar
  • 73
3 votes
0 answers
896 views

How does one compute the Chern classes of the dual sheaf from the Chern class of the original sheaf?

Let $X$ be a smooth projective 4-fold (over $\mathbb{C}$). Let $Z \subset X$ be a codimension two subscheme. Let $I_{Z}$ denote the ideal sheaf of $Z$. How does one compute the Chern classes of $I_Z^{...
Ritwik's user avatar
  • 3,235
4 votes
1 answer
158 views

Rationality of $V_1$ fano threefold

In the book of Iskovskikh and Prokhorov it seems not known wether the $V_1$, an hypersurface of degree $6$ in the weighted projective space $\mathbb{P}(3,2,1,1,1)$, is rational or not. Is there any ...
Xavier Roulleau's user avatar
3 votes
1 answer
331 views

uniqueness of uniformizers

Let R be a noetherian normal domain (if it makes any difference, I'm happy to assume R is also local). If $p$ is a height one prime, then the localization $R_p$ is a dvr, hence the maximal ideal $...
Yosemite Sam's user avatar
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1 vote
1 answer
287 views

Homological dimension of pure coherent sheaves and specialization

Let $X$ be a projective variety, not necessarily smooth, $R$ a DVR with residue field $k$ (assume char$(k)=0$). I am looking for examples of a pure coherent sheaf, say $F$, on $X_R:=X \times_k \mathrm{...
user45397's user avatar
  • 2,195
3 votes
0 answers
263 views

Exterior tensor of derived categories of coherent sheaves

Let $X, Y$ be Noetherian perfect derived stacks over $S$ a regular perfect derived stack. Consider the exterior tensor functor $$\text{DCoh}(X) \otimes_{\text{DCoh}(S)} \text{DCoh}(Y) \rightarrow \...
Harrison Chen's user avatar
3 votes
1 answer
556 views

Comparing the definitions of perfect complexes on algebraic stacks and schemes

Let $X$ be an algebraic stack. In Perfect complexes on algebraic stacks (4.1) a perfect complex $P \in \mathsf{D}_{\mathrm qc}(X)$ is defined to be a complex such that, for any smooth morphism $\...
Francesco Genovese's user avatar
3 votes
0 answers
173 views

Cohomology classes functorial under etale morphisms

Consider the full subcategory $\mathcal C$ in the big etale site of a field $k$ consisting of smooth schemes, namely the category of smooth varieties over a field $k$ with etale maps as morphisms. I ...
SashaP's user avatar
  • 7,027
0 votes
2 answers
2k views

Tensor products of two domains

Let $R$ be an integral noetherian regular local ring. Let $S$ be a noetherian integral domain such that $S/R$ is finite. That is, $R \subset S$ and the surjection $R^{\oplus n} \twoheadrightarrow S$ ...
Pierre MATSUMI's user avatar
9 votes
3 answers
1k views

Klein's curve (algebraic geometry)

I can't find any information about the canonical ring of Klein's quartic curve (the one with 168 automorphisms). I would imagine there is a lot known about the structure of this ring. Can anybody help ...
Emma Previato's user avatar

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