Questions tagged [ag.algebraic-geometry]
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
21,607
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Configuration of the branch locus of a branched covering of an elliptic curve
Let $C$ be a curve of genus 3 and suppose that it admits a branched cover $\varphi:C\rightarrow E$ with $E$ elliptic and such that $\varphi$ does not factor through any \' etale cover. Then the degree ...
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3
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Algorithms in Invariant Theory
Let $V$ be a polynomial representation of the general linear group $\Gamma:=\DeclareMathOperator{\Gl}{Gl}\Gl_n(\newcommand{\C}{\mathbb C}\C)$.
In chapter 4.6 of his book "Algorithms in Invariant ...
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Why are Gromov-Witten invariants of K3 surfaces trivial?
Why is GW invariants of K3 surfaces are trivial? My naive guess is that GW invariants are deformation invariant and you can always deform your K3 surface to non-projective one, which has no subcomplex ...
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1
answer
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reduction of elliptic curves
Let $X$ be an elliptic curve over a complete local field.
The definition of semi-abelian reduction is: "the Neron model of $X$ is a semi-abelian scheme". On the other hand, the definition of semi-...
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6
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Characteristic zero and characteristic $p$ in algebraic geometry
Are there non-trivial (i.e. excluding concepts that can be defined only for $p>0$) statements in algebraic geometry that hold for all fields of characteristic $p$ for all prime $p$ but are known ...
user16974
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1
answer
841
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$\mathcal{D}$-modules of level m
My question is regarding the definition of $\mathcal{D}$-modules of level $m$ given in this paper. As an example, let $X=\mathbb{A}^1$ over $S=\text{Spec }\overline{\mathbb{F}_p}$; I was told that a $\...
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Lefschetz fixed point formula: an 'easy' proof; cohomology with compact support
I have two questions concerning the LFPF (for etale cohomology).
Is there an easy 'explanation' of this statements (that could be understood by students)? In particular, I would like to get away with ...
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votes
2
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Examples of Calabi-Yau that are birational to each other?
I was told that Calabi-Yau's can be birational to each other but not isomorphic (biholomorphic).
But I've never seen explicit examples. Can anybody here show me one?
(E.g. maybe an explicit ...
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1
answer
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Serre duality and Hirzebruch-Riemann-Roch in the non-projective case
Serre duality and the Hirzebruch-Riemann-Roch formula are usually stated for $X$ a smooth projective algebraic variety. Do you know of a reference which proves these results for $X$ smooth and proper?
...
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Deformation of space curves to union of lines
Does every irreducible component of a Hilbert scheme of curves in $\mathbb{P}^3$ contain a curve that is a union of lines (not necessarily reduced)? Furthermore, given a curve $C \subset \mathbb{P}^3$ ...
- 1,020
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votes
2
answers
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What's the meaning of pencils in birational geometry?
I see in some books the authors call a one dimensional linear system a pencil, but in other books one call a linear system $|D|$ is not compsited with a pencil if $\dim \phi_{|D|}(X)\geq 2$ and even ...
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Can flatness be specified by a natural coherent sheaf?
More precisely:
Given a finite-type morphism $f \colon X \to Y$ of nice schemes (say, both of finite type over a field), is there a "natural" coherent sheaf $\mathcal F_f$ on $X$ such that the ...
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votes
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Hurwitz numbers and Hurwitz theory
Am currently doing a self-study of hurwitz numbers and Hurwitz theory, is there a good source for one with only basic knowledge of algebraic geometry. Notes and papers are also welcome.
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Exceptional sheaves and double dual
In his beautiful paper Moduli of bundles on $K_3$ surfaces, Mukai proves that if $F$ is a rigid (i.e. $\mathrm{Ext}^1(F,F) = 0$) torsion free sheaf on a surface $S$ with $|-K_S| \neq 0$, then $F^{**}$,...
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When do blowups ''commute''?
Let $M$ be a manifold (variety, scheme, your favorite object) and let $N_1,N_2$ be two submanifolds (subvarieties, closed subschemes, ideal sheafes, etc.) such that $N_1 \cap N_2 \neq \emptyset$. ...
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When is an orbit spherical?
I asked the following question over at math.stackexchange, but got no answers. Maybe it's less well-known than I thought, but I still wanted to ask here:
Let's assume we have an affine, reductive, ...
- 4,822
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Cohen-Macaulay sheaves which are not locally free
A coherent sheaf $\mathcal{F}$ over a Noetherian scheme $X$ is called (maximal) Cohen-Macaulay if $depth_{\mathcal{O}_x}(\mathcal{F}_x) = \dim\mathcal{O}_x$ for any $x\in X$, where $\mathcal{O}_x$ is ...
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2
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Du Val Singularities and Dynkin diagrams references
May I ask whether there are good references for computing blowups of the Du Val Singularities? Also, how are these singularities related to the Dynkin diagrams?
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Philosophy behind Mochizuki's work on the ABC conjecture
Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy ...
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Why is the base of SLAG fibration of CY3 expected to be $S^3$?
The SYZ conjecture roughly says that any Calabi-Yau threefold $X$ has a special Lagrnagian fibration $\pi:X\rightarrow B$ by 3-tous with section and one of its mirror partners $\check{X}$ is obtained ...
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Partitions of $\mathbb{R}^d$ by implicit polynomial equations
Given a polynomial
$p(x_1,x_2,\ldots,x_d)$
in $d$ variables, with maximum degree $k$,
what is the maximum number of
components of $\mathbb{R}^d$ minus $p(\ldots)=0$?
In other words, into how many ...
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Lift of a morphism between geometric quotients
Let $S$ be a scheme.
Definition. Let $X$ be an $S$-scheme and $G$ a smooth affine group $S$-scheme acting on $X.$ An $S$-scheme $Y$ is a geometric quotient of $X$ by $G$ if there exists a morphism $\...
user17778
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Image of linear projection
Let $X \subset \mathbb{P}^n$ be a projective variety (i.e. zariski closed), and let $\pi : X \dashrightarrow \mathbb{P}^m$ a linear projection ($\pi$ is not in every point of $X$ defined).
Under ...
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Leray degeneration for smooth projective morphisms and formality of families of compact Kähler manifolds
Let $\pi \colon X \to S$ be a smooth projective morphism of algebraic varieties, say over $\mathbf C$. By Deligne's argument ("Théorème de Lefschetz...", 1968), there is for each $i$ an injection
$$ \...
- 39.3k
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votes
1
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Rational equivalence and Hilbert flag scheme
Given a smooth surface $X \subset \mathbb{P}^3$ if we have two curves $C_1, C_2$ that are rationally equivalent is it true that both $(C_1,X)$ and $(C_2,X)$ will be in the same irreducible component ...
- 1,020
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votes
2
answers
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Blowing down -1 curves
After a good look for anything in the way of an explicit example, and harassing the algebraic geometers in the department, I still am unable to find an answer to my question, so any light will be very ...
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1
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local galois representation with higher coefficient
Suppose K is a local field , G is its galois group, V a fine dimensional Vector space over F, which is a sub field of K, and totally ramified over $Q_p$. Consdider the linear action of G on V (V is ...
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$T^2$-fibered K3 surface with involution
Let $S$ be a K3 surface and $f:S\rightarrow \mathbb{P}^1$ a $T^2$-fibration (not necessarily holomorphic, I have a special Langrangian fibration in mind). Assume there is a $k$-section, then a fiber ...
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5
votes
1
answer
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Bruhat decomposition of a quadric hypersurface
On page 3 of this paper, the authors give a Bruhat cell decomposition of a quadric hypersurface $Q$ of complex dimension $n$. This may be a stupid question, but it doesn't seem clear to me what ...
8
votes
1
answer
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Whether Hilbert schemes of 3 points on arbitrary smooth projective varieties are smooth
As is well known, the Hilbert scheme of two points on a given smooth projective variety X are blow up along diagonal of product of X and then quotient the Z2 action. It is smooth.
My question is ...
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0
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Weak admissibility in algebraic families
Let $M$ be an algebraic family of isocrystals over a base scheme $R/\mathbb{Q}_p$ (not a rigid analytic space).
The question is: is the set of weakly admissible points (i.e., the points $r\in R$ ...
- 1,831
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Non-compact Kähler manifolds which admit a positive line bundle
A complex manifold which admits a positive line bundle is automatically Kähler. Furthermore, if the manifold is compact, then it is projective by the Kodaira Embedding Theorem. In particular, not ...
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Contracting Fano divisors
Suppose we are given a smooth complex algebraic variety $X$ with a smooth, irreducible divisor $D$ such that $D$ is Fano and the normal bundle $L$ to $D$ is anti-ample. Then we can contract $D$ to a ...
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Can a birational morphism between smooth varieties be dominated by smooth blowup sequences?
Suppose $f:X\rightarrow Y$ is a birational morphism between smooth varieties and D is a snc divisor on $Y$. Can we find a smooth blowup sequence on $Y$ which dominates $f$ such that the preimage of D ...
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0
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On the structure of commutative group schemes
The structure of a commutative affine algebraic group $G$ over a field $k$ is understood (SGA 3): $G$ has a maximal subgroup of multiplicative type $M$ and the quotient $G/M$ is unipotent.
I am ...
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votes
1
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About the strength of representation-theoretic obstructions for orbit closure problems
Let $G$ be a reductive, affine, algebraic group over $\newcommand{\C}{\mathbb C}\C$. Let $X$ be a $G$-variety. For $x\in X$, we write
$$G_x:=\{ g\in G\mid g.x=x\}$$
for its stabilizer and for any ...
- 4,822
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Singular fibers of an elliptic fibered K3 surface.
Let $f:S\rightarrow \mathbb{P}^1$ be an elliptic K3 surface. Assume that $\mathrm{Pic}(S)\cong U$, where $U$ stands for the hyperbolic lattice. I think that the elliptic fibration has only singular ...
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1
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1
answer
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Homological equivalence for algebraic cycles
How does one prove that for a smooth projective variety $X$ over an algebraically closed field $k$, an algebraic cycle that is homologically equivalent to zero is also numerically equivalent to zero? ...
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1
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Chern classes of ideal sheaf of an analytic subset
Let $X$ be a Kähler manifold of dimension $n$, and $Z \subset X$ an analytic subset of codimension $k$. I have read in a paper the following result, a proof of which I cannot find:
$$c_k(\mathscr{I}...
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1
answer
930
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Complex structures on a K3 surface as a hyperkähler manifold
A hyperkähler manifold is a Riemannian manifold of real dimension $4k$ and holonomy group contained in $Sp(k)$. It is known that every hyperkähler manifold has a $2$-sphere $S^{2}$ of complex ...
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How to get Haar measure on a compact Lie group, given the complexification?
This is the first in what may be a series of questions on the theme "a Banach algebraist/Bear Of Little Brain needs help with algebraic geometry".
$\newcommand{\Cplx}{{\mathbb C}}\newcommand{\fg}{{\...
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Cup products and hypercohomology
This is a cross-post of the following math.stackexchange question: https://math.stackexchange.com/questions/188760/cup-product-and-hypercohomology
I always found the cup product slightly mysterious. ...
- 1,951
15
votes
1
answer
892
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Monsky's proof of the finiteness of de Rham cohomology
I'd really like to understand the proof that Paul Monsky wrote about the finiteness of the de Rham cohomology of algebraic varieties. I'd like it very much because it seems to explain in concrete ...
- 1,044
8
votes
1
answer
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When do blow-up and quotient commute?
Let a finite group $G\subset SL(n,\mathbb{C})$ act on $\mathbb{C}^{n}$ in a natural way. Assume there is a crepant resolution of $f:X\rightarrow \mathbb{C}^{n}/G$. When is it possible to write $X$ as $...
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answer
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Second lowest weight piece of the cohomology of an algebraic variety
If $U$ is a smooth algebraic variety, then one can give a simple description of the lowest weight part of its cohomology: if $X$ is a smooth compactification and $j \colon U \to X$ the inclusion, then ...
- 39.3k
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635
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Algebraicity of power series over the rationals from the algebraicity over Fp
Van der Poorten conjectures [in "Power series representing algebraic functions," Sem. Th. Nombres Paris 1990-91] that if a power series over the rationals is the [complete] diagonal [of a rational ...
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2
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Algebraic curve mapping to elliptic curve - how to check whether this is possible?
Question: Let $C$ be an algebraic curve over some field (like the rationals) given by a plane projective model (possibly with singularities). Is there an easy way to see if this curve has a non-...
- 20.8k
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votes
1
answer
595
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Semiorthogonal decompositions for Fano 3-folds and 4folds
Let $X$ be a projective Fano 3-fold or 4-fold and let $D^b(X)$ be the bounded derived category of coherent sheaves on $X$. For what $X$ is it known a semi orthogonal decomposition into indecomposable ...
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votes
0
answers
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Randomized alternative to Buchberger's algorithm
Richard Lipton's blog describes a A New Way To Solve Linear Equations by Prasad Raghavendra.
Can the ideas in this algorithm be generalized to systems of polynomial equations to provide a randomized ...
- 529
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votes
3
answers
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Is SL(2,C)/SL(2,Z) a quasi-projective variety?
Consider the complex 3-fold $SL(2,\mathbb C)/SL(2,\mathbb Z)$ (just for clarity: note that $SL(2,\mathbb Z)$ acts without stabilizers, so this is a complex manifold, not a complex orbifold).
Is $SL(...
- 18.3k