Questions tagged [ag.algebraic-geometry]
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
21,606
questions
5
votes
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Admissible subcategories of $D^b(\mathbb{P}^n)$
Recall that a triangulated subcategory $\mathcal{A}$ of a triangulated category $\mathcal{B}$ is called admissible if the inclusion functor has both left and right adjoints.
Is it true that all ...
- 10.5k
4
votes
1
answer
1k
views
Dimension of the global sections of the Serre twisting sheaf on a curve
Let, $C$ be a projective curve (not necessarily reduced), $i:C \to \mathbb{P}^n$ be a closed immersion. Does there exist a bound on/geometric interpretation of the dimension of $H^0(i^*(\mathcal{O}_{\...
- 1,573
3
votes
0
answers
411
views
Lines on Fano complete intersections
Let $X \subset \mathbb{P}^n$ be a non-singular complete intersection of $s$ hypersurfaces
of degrees $d_1,\dots,d_s$ over an algebraically closed field $k$ of characteristic zero. Let $d=d_1 + \dots + ...
- 20.8k
0
votes
1
answer
141
views
meaning of $k(C)/1+\mathfrak{m}_x$ [closed]
Let $C$ be a smooth projective curve over some field $k$ and $x$ a closed point of $C$. I've seen some constructions in which people use
$k(C)^\times / 1+\mathfrak{m}_x$.
What's the meaning of that?...
7
votes
1
answer
704
views
Some special complex tori
Let $T=V/\Gamma $ be a complex torus; so $V$ is a finite-dimensional complex vector space and $\Gamma$ a lattice in $V$.
Moreover I have a positive definite hermitian form $H$ on $V$ such that the ...
- 37.3k
1
vote
1
answer
251
views
Moduli spaces admitting birational morphisms over moduli spaces of curves
There are many alternative compactifications of $M_{g,n}$ which live naturally under the classical Deligne-Mumford compactification $\overline{M}_{g,n}$.
For instace the moduli spaces of weighted ...
- 8,852
3
votes
1
answer
239
views
Etale covers of products of curves
Is a finite etale cover of a product of curves again a product of curves?
The answer is no in general. Here's one way to construct an example. Take the product $A$ of two elliptic curves and an ...
- 31
2
votes
2
answers
1k
views
singularities of the dual variety of a surface
I am looking for a proof/reference of the following simple fact, which I think it holds true.
Let $S\subset \mathbb{P}^n$ be a surface embedded by a very ample linear system. Then I know that the ...
- 3,717
0
votes
1
answer
267
views
Intuition on an object in algebraic geometry
Let $X$ be a smooth projective variety over a field $k$ of characteristic zero, and let $D$
be a simple normal crossing divisor inside $X$ having irreducible components $D_i$.
Further let $x \in X$ be ...
0
votes
1
answer
225
views
Can generalization of Mumford’s theorem imply Mumford’s theorem for surface?
Mumford’s theorem for surface says that for a surface $S$ with $p_g(S)\neq0$ ,$\text{CH}_0(S)$ is not representable(or infinite-dimensional). But in Voisin's LECTURES ON THE HODGE ANDGROTHENDIECK–...
- 73
4
votes
1
answer
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Any non-constant surjective holomorphic map between connected compact complex manifolds of equal dimension is a ramified finite-sheet covering
I'm reading this site:holomorphy of inverse map
There is a statement made by Colin Tan at the last answer made by himself.
Any non-constant surjective holomorphic map between connected compact ...
- 41
4
votes
1
answer
1k
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on two definitions of irreducible connection
I have seen two kinds of definitions of irreducible connections on fibre bundls: A connection is said to be irreducible if
the holonomy group is precisely $G$ and not a proper subgroup.
or
2. there ...
- 41
11
votes
1
answer
665
views
Pro-algebraic versus continuous Galois cohomology, and schematic homotopy types
I've been thinking about Bertrand Toen's approach to studying the homotopy theory of schemes, and I've come across an inconsistency in my understanding of the subject that I was hoping somebody might ...
- 1,818
2
votes
0
answers
108
views
Changing the Hilbert scheme of curves by adding the hyperplane section
Let $X$ be a smooth degree $d$ ($d>4$) surface in $\mathbb{P}^3$.
Let $C$ be a divisor on $X$. Let $D$ be a general element of the linear series $|C+H_X|$ where $H_X$ is the hyperplane section of $...
- 1,020
0
votes
1
answer
129
views
spectrum of an induced algebra
Let $G$ be a reductive group defined over an algebraically closed field $k$ and $B$ be a fixed Borel subgroup of $G$. Suppose $X=Spec(R)$ is an affine scheme with $B$ rationally acting on it; hence $B$...
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3
votes
2
answers
484
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Varieties that do not extend to flat families
Is it easy to give an example of a function field $K$ and a smooth proper variety $X$ over $K$ that does not extend to a flat scheme over $B$, where $B$ is a smooth proper variety with function field $...
- 31
5
votes
2
answers
730
views
Degree of a projective scheme and its defining equations
Let $X \subset \mathbb{P}^n$ be any projective scheme. Denote by $I_X$ the (saturated) ideal of $X$. Suppose the degree of $X$ is $d$. Under what assumptions there exists a polynomial in $I_X$ of ...
- 2,022
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votes
2
answers
893
views
Is there an algorithm to compute efficiently the dessin d'enfant from a Belyi pair?
Let $(X,f)$ be a Belyi pair, i.e. a Riemann surface $X$ together with a morphism $f: X \to \mathbb{P}^1$, ramified only in $0,1, \infty$. Grothendieck's dessin d'enfant is the pre-image $G$ of the ...
- 4,060
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votes
1
answer
994
views
Derived categories of arithmetic schemes?
Let $X$ be a smooth scheme of finite type over $\mathbb Q$ or $\mathbb Z$. It is natural to consider $D(X)$, the derived category of coherent sheaves on $X$. Beyond the definition, have there been any ...
- 5,166
1
vote
1
answer
192
views
Picard number of fundamental divisor of Fano 3-fold
Given a (smooth)Fano 3-fold $X$, Sokurov proved that the fundamental linear system contains a smooth surface. My question is :
If the Picard number of X is 1,Is there such a smooth surface(in the ...
- 1,072
0
votes
1
answer
541
views
Canonical bundle of the moduli space of curves
By the pointed canonical bundle formula the canonical bundle of $\overline{M}_{g,n}$ is given by
$$K_{\overline{M}_{g,n}} = 13\lambda+\psi-2\delta-\sum_{I}\delta_{1,I}$$
where $\lambda$ is the Hodge ...
- 8,852
0
votes
1
answer
313
views
Symplectic structure on $Sym^kG^{\mathbb{C}} $
Let $G$ be a compact Lie group, and let $G^\mathbb{C}$ be its complexification.
I am looking for a symplectic structure (without use of coordinates) on
$$
Sym^kG^{\mathbb{C}},
$$
PS:Here $G^{\mathbb{...
user21574
2
votes
1
answer
646
views
Kodaira dimension of the moduli space of curves
It is known that the moduli space $\overline{M}_{g}$ of genus $g$ curves is of general type for $g\geq 24$.
By Theorem 2.4 of
Logan, Adam The Kodaira dimension of moduli spaces of curves with ...
- 8,852
2
votes
2
answers
584
views
Fixed point for a self-mapping on subset of C[0,1]
Let $f_1$ and $f_2$ be arbitrary self-mappings on $C([0,1])$ with $f_2 > f_1$. Define set $F = \{f \in (C[0,1])| f_1 \leq f \leq f_2 \mbox{ and } f \mbox{ is increasing}\}$. Is it true that every ...
- 23
1
vote
1
answer
259
views
Chow groups of finite covers of unirational varieties
Suppose $f:X \to Y$ is a finite morphism with $X$ and $Y$ being affine varieties, such that $X$ is unirational. In fact $X$ is more than unirational, it is the image of a morphism from a zariski open ...
- 327
3
votes
0
answers
175
views
Existence of a faithfully flat algebra with a cancellation property
Let $X$ be a scheme (or even algebraic stack). Let's assume that it is nice enough, for example noetherian.
Is there a (faithfully flat) $\mathcal{O}_X$-algebra $\mathcal{A}$ with the following ...
- 61.5k
2
votes
1
answer
248
views
Computing Ext: $\text{Ext}(i_* \mathcal{O}_X, i_* \mathcal{O}_X)$ for closed embedding $i:X \rightarrow Y$
Let $V$ be a vector bundle on $X$, and $Y = \text{Tot}(V)$ be the total space of this bundle; we have a closed embedding $i: X \rightarrow Y$. Why is the following result true?
$$ \text{Ext}^k(i_* \...
- 21
2
votes
2
answers
273
views
Segre class of smooth vector bundles over smooth manifolds?
Before you read the following question, please assume I have no knowledge in algebraic geometry.
Is it possible to define Segre class of a smooth complex vector bundle over a smooth manifold by using ...
- 1,087
0
votes
1
answer
394
views
Intersection of lines and a closed variety in projective space
Suppose $X$ is a proper closed subset of $\mathbb{P}^n_k$, $P$ a point not in $X$ and a hyperplane $H$. Denote $m_{P,H}$ the supremum of intersections of straight lines from $P$ to points in $H$ with $...
user39380
3
votes
0
answers
419
views
Ext groups of affine scheme
Let $A$ be a commutative ring. $\textbf{Spec}(A)$ is the the
spectrum of $A$. $M$ and $N$ are $A$-modules. $\tilde{M}$ and
$\tilde{N}$ are sheaves associated on $\textbf{Spec}(A)$
respectively.
I ...
- 481
0
votes
2
answers
2k
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global sections of canonical line bundle of a projective variety
Given a smooth projective variety $X \subset \Bbb{CP}^k$ why is it true that global sections of $O(l)|_X, l >> 0 $ are just global sections of $O(l)$ on $\Bbb{CP}^k$ restricted to $X$?
Here $...
- 1,171
4
votes
3
answers
2k
views
transcendence of periods of CM elliptic curves
Let $E$ be an elliptic curve over $\overline{\mathbb{Q}}$ defined by a Weierstrass equation
$$
y^2=4x^3+g_2x+g_3.
$$ Then $H^1_{dR}(E/\overline{\mathbb{Q}})$ is spanned by the classes of the ...
- 49
3
votes
1
answer
633
views
looking for an identity for higher jet bundle $J^kM$?
We know this fact that the first jet Bundle $J^1M$ is diffeomorphic with $T^*M×\mathbb{R}$.i.e,
($J^1M=T^*M×\mathbb{R}$)
Is there something like this identity for higher jet bundle $J^kM$?
I editted ...
user21574
9
votes
2
answers
829
views
$Pic$ of the stack of elliptic curves vs. $Pic$ of the coarse space
There's a natural map $f:\overline{\mathcal{M}}_{1,1}\to \overline{M}_{1,1}\cong \mathbb{P}^1$ from the stack of elliptic curves to the coarse space. Both spaces have $Pic=\mathbb{Z}$ hence $f^*:\...
- 3,717
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votes
3
answers
2k
views
How many integer points does my favorite ellipse go through?
The equation of the ellipse interpolating the six lattice points $(0,0)$, $(1,0)$, $(0,1)$, $(d-1,d)$, $(d,d)$, $(d,d-1)$ in the plane for a fixed $d$ (at least 3) is
$$
x^2+y^2 - \frac{2(d-1)}{d}xy-x-...
- 174
1
vote
0
answers
270
views
stably birational abelian varieties are isomorphic
Can anybody help me to prove the following result:
Proposition. Let $A$ and $B$ be abelian varieties over a field $k$ of characteristic zero. Assume that $A \times \mathbb{P}_k^n$ and $B \times \...
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3
votes
0
answers
252
views
How do I compute the Azumaya locus?
I have an algebra $A$ that is finite over its centre $Z(A)$ and I want to compute the Azumaya locus of $Z(A)$ (or equivalently, its ramification locus).
Looking in McConnell-Robson Noncommutative ...
- 1,831
2
votes
1
answer
101
views
Which actions preserve non-complete intersections?
Let $X$ be a smooth projective variety and $Z$ is a closed subscheme in $X$ which is not a complete intersection in $X$. Assume the dimension of $X$ (resp. $Z$) is greater than $3$ (resp. $1$). Then,
...
- 2,022
0
votes
2
answers
620
views
Rationality of curve does not depend on base change
By a curve I mean an integral one-dimensional scheme of finite type over a spectrum of a field.
Let $C$ be a curve over an arbitrary field $k$. It's probably a very well known fact, that $C$ is ...
- 267
3
votes
1
answer
293
views
Are weak Fano 4-folds with canonical Gorenstein singularities bounded?
A Fano variety over $\mathbb{C}$ with Gorenstein singularity is called weak Fano if the anti-canonical divisor is nef and big.
Are there finite families of weak Fano 4-folds with canonical ...
- 3,362
5
votes
1
answer
338
views
Looking for a reference (on GW invariants of quintic)
1) Apparently, physicist can calculate the GW invariants of quintic CY 3-fold up to genus 51.
I am looking for a reference that has a table of these number for some low degrees (say up to degree 5) ...
5
votes
1
answer
990
views
Bruhat decomposition for reductive groups in characteristic zero?
Let $G$ be a reductive, linear algebraic group (variety) over an algebraically closed field $\Bbbk$ of characteristic zero. If $G$ is connected, I know from Humphrey's book that for any Borel subgroup ...
- 4,822
2
votes
1
answer
587
views
Dual of a Complex 2-Torus
Is a complex torus $A$ of dimension 2 always isomorphic to its dual torus (i.e. the torus obtained by taking the dual lattice), or are there counterexamples to this?
- 761
4
votes
2
answers
701
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on Brieskorn Manifolds
Brieskorn showed that $X_{k}=\{(z_0, \dots, z_k)\in
\mathbb{C}^{k+1}| -z_{0}^{3}+\sum_{i=1}^{k} z_{i}^{2}=0\}$(k odd, k>2) is a
topological manifold. Is it a smooth manifold?
In general, let $a_1, \...
- 481
5
votes
2
answers
643
views
Action of automorphisms of a $K3$ surface on its $(-2)$-curves
Consider a complex $K3$ surface $X$ and take its group of automorphisms $Aut(X)$. It is a known fact that the action of $Aut(X)$ on the set of rational $-2$ curves of $X$ has only finite number of ...
- 14k
4
votes
1
answer
636
views
Abel-Jacobi map
Let $X$ be a smooth projective variety defined over a number field $F$ and consider the Abel-Jacobi map $\mathrm{AJ}_k:\mathrm{CH}_0^k(X_{\overline{\mathbb{Q}}})\rightarrow \mathrm{Jac}^{2k-1}(X)$, ...
- 273
3
votes
1
answer
209
views
Hodge numbers of symmetric squares
Let $X$ be a projective variety.
Consider $Sym^2X$, the quotient of $X \times X$ by the involution $(x, x') \mapsto (x', x)$.
What is the relation between the (mixed) Hodge numbers of $Sym^2 X$ ...
- 31
4
votes
1
answer
1k
views
Can motivic E_∞-ring spectra be strictified to commutative motivic symmetric ring spectra?
Theorem 4.5.4.7 (4.4.4.7 in the old version) in Lurie's Higher Algebra (or Theorem 4.3.22 in DAG III) states (roughly speaking) that under certain conditions
the ∞-category of commutative ∞-monoids in ...
- 36.3k
4
votes
1
answer
413
views
Automorphicity of L-Factors of Zeta Functions
Associated to a variety over a number field $K$, one has a family of "Hasse-Weil" L-functions, which can be combined (as an alternating product) to give the Hasse--Weil zeta function of the variety, ...
- 643
3
votes
2
answers
1k
views
Homological smoothness implies projectivity?
Let $A$ be a unital associative algebra over a commutative noetherian ring $R$. Assume that $A$ is homologically smooth, which means that $A\in D_{perf}(A\otimes_R^L A^{op})$, which also means that $A$...
- 1,942