Questions tagged [ag.algebraic-geometry]
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
21,545
questions
2
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437
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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 ...
7
votes
1
answer
391
views
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 $\rho: \pi_1(U, u) \to \text{Sp}(E/(E \cap E^\perp), \psi)$ ...
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 $...
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 ...
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 ...
10
votes
0
answers
865
views
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, ...
2
votes
1
answer
296
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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 ...
1
vote
1
answer
104
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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,\...
1
vote
1
answer
228
views
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 ...
6
votes
1
answer
179
views
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 ...
6
votes
1
answer
1k
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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.
5
votes
1
answer
299
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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 ...
4
votes
1
answer
916
views
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\...
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 ...
4
votes
1
answer
357
views
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}(...
0
votes
2
answers
391
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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$, ...
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 ...
2
votes
0
answers
73
views
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 ...
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 ...
1
vote
0
answers
99
views
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 ...
3
votes
1
answer
411
views
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 ...
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) ...
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
$$
over a finite field $k = \mathbb{F}_{2^d}$, $\mathrm{gcd}(d, 6)...
3
votes
0
answers
131
views
Arithmetic version of "Attaching maps" for moduli of curves
I am looking for a reference for attaching maps of moduli of curves with marked points. Especially I would like to know whether they descend over $\mathbb{Z}$. On one hand this seems very hard to ...
3
votes
1
answer
540
views
Dimension of the zero weight space in $V_{2\rho}$
Let $\rho$ be the half sum of the positive roots (also the sum of fundmental weights) for $SL_n(\mathbb C)$. Then $2ρ$ is in the root lattice. Then what is the dimension of the zero weight space for ...
1
vote
1
answer
151
views
Nature of morphism between Brill Noether varieties
Let $C$ be a smooth curve over complex numbers. Consider the Brill Noether varieties. If $g$ is the genus of $C$. If $r,d$ are positive integers,
$$W^r_d=\{A\in Pic^d(C): h^0(A)\geq r+1\},$$
$$G^r_d=\...
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 ...
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 ...
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 $0$....
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 ...
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 ...
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 ...
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 ...
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, ...
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 ...
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 ...
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?
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 ...
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:
$$
0 \rightarrow \mathbb{Z} \rightarrow \mathscr{O}_X \rightarrow \mathscr{O}_X^\times \...
10
votes
1
answer
496
views
Does GAGA hold over other topological fields?
If k is a non-discrete topological field, we can define an analytic space over k just like complex analytic spaces over $\mathbb{C}$. If you replace "complex analytic space" and "complex algebraic ...
1
vote
0
answers
196
views
(Ordered) Configuration space in algebraic geometry
Let $X$ be a topological space and denote by $F_n(X)$ the following subspace:
$$F_n(X):=\{(x_1,\cdots ,x_n)\in X^n: x_i\neq x_j \forall i\neq j\}.$$
Note that, we are not considering the quotient of $...
3
votes
0
answers
896
views
How does one compute the Chern classes of the dual sheaf from the Chern class of the original sheaf?
Let $X$ be a smooth projective 4-fold (over $\mathbb{C}$). Let $Z \subset X$ be a codimension two subscheme. Let $I_{Z}$ denote the ideal sheaf of
$Z$. How does one compute the Chern classes of $I_Z^{...
4
votes
1
answer
158
views
Rationality of $V_1$ fano threefold
In the book of Iskovskikh and Prokhorov it seems not known wether the $V_1$, an hypersurface of degree $6$ in the weighted projective space $\mathbb{P}(3,2,1,1,1)$, is rational or not. Is there any ...
3
votes
1
answer
331
views
uniqueness of uniformizers
Let R be a noetherian normal domain (if it makes any difference, I'm happy to assume R is also local).
If $p$ is a height one prime, then the localization $R_p$ is a dvr, hence the maximal ideal $...
1
vote
1
answer
287
views
Homological dimension of pure coherent sheaves and specialization
Let $X$ be a projective variety, not necessarily smooth, $R$ a DVR with residue field $k$ (assume char$(k)=0$). I am looking for examples of a pure coherent sheaf, say $F$, on $X_R:=X \times_k \mathrm{...
3
votes
0
answers
263
views
Exterior tensor of derived categories of coherent sheaves
Let $X, Y$ be Noetherian perfect derived stacks over $S$ a regular perfect derived stack. Consider the exterior tensor functor
$$\text{DCoh}(X) \otimes_{\text{DCoh}(S)} \text{DCoh}(Y) \rightarrow \...
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 $\...
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 ...
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 $R^{\oplus n} \twoheadrightarrow S$ ...
9
votes
3
answers
1k
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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 ...