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

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3
votes
1answer
230 views

When a smooth algebra is regular?

Let $A \subseteq B$ be noetherian integral domains, $A$ regular (=every localization at maximal ideal is a regular local ring) and $B$ is a smooth $A$-algebra. For the definition of a smooth algebra, ...
3
votes
0answers
255 views

What's the name of this branched covering?

I've come across a double cover of $\mathbb P_1(\mathbb C)$, ramified at $[1:1]$ and $[-1:1]$ in homogeneous coordinates, given as the quotient by the natural $\mathbb Z/2\mathbb Z$-action generated ...
2
votes
1answer
158 views

On the number of irreducible components of an exceptional divisor

Let $X$ be a complex, affine variety and $Z\subseteq X$ a closed subset of $X$ (i.e. a closed, reduced subscheme). Let $E$ be the exceptional divisor of the blow-up $\pi:\tilde X\to X$ of $X$ with ...
1
vote
1answer
135 views

Can a rigid CY threefold have infinitely many automorphisms

Let $X$ be a rigid Calabi-Yau threefold. Does $X$ have only finitely many automorphisms? N.B. A smooth projective threefold $X$ over $\mathbb C$ is a rigid Calabi-Yau variety if $h^i(X,\mathcal O_X) ...
1
vote
0answers
154 views

Fiber Bundle with a perfect bilinear map

Let $\pi:X\rightarrow Y$ a double cover of Riemann surfaces, and consider a $SL_n-$bundle $E$ with a perfect $\mathcal O_Y-$bilinear pairing $$\psi:\pi_*E\times\pi_*E\rightarrow\pi_*\mathcal O_X$$ ...
7
votes
0answers
77 views

Equation of the curve corresponding to a polarization of an abelian surface

Let $\mathbb{C}^2/\Lambda$ be a polarized abelian surface. I think it is well-known how to write down the equation of the divisor corresponding to the polarization, in terms of theta functions etc. ...
3
votes
1answer
121 views

Enriques classification of algebraic surfaces in characteristic zero

I am searching for a reference about the classification of algebraic surfaces over an arbitrary algebraically closed field of characteristic zero. In the 1949 book "le superficie algebriche" by ...
7
votes
1answer
186 views

Is every proper regular relative algebraic space curve over a Dedekind domain projective?

This question is in some sense a follow up to a related question Is a normal proper relative curve over a DVR projective? Let $R$ be a Dedekind domain, let $S := \mathrm{Spec}(R)$, and let $X ...
4
votes
2answers
199 views

Secant varieties of curves in $\mathbb{P}^4$

My question is motivated by the following simple observations. By a standard dimensions count in $\mathbb{P}^4$ there should not exist neither an hypersurface of degree $3$ with multiplicity $2$ in ...
3
votes
2answers
188 views

Do discrete valuation rings correspond to local rings of points in fibre?

Given projective curves $C$ and $C'$ and a surjective morphism $\varphi\colon C\to C'$, such that $Q\in C'$ is a smooth point and its fibre $\varphi^{-1}(Q)$ consists of smooth points. Then ...
8
votes
1answer
509 views

Pros and cons of Stacks Project as a reference compared with EGA/SGA

I would like to know pros and cons of Stacks Project compared with EGA and SGA and whether it serves as a nice alternative to them. Since I haven't read both of these texts, my attempt to compare the ...
6
votes
2answers
178 views

Singularities of Pfaffian hypersurfaces

Let $X\subset\mathbb{P}^4$ be an hypersurface of degree six given by the Pfaffian of a $6\times 6$ matrix $M$ whose entries are quadratic forms in the homogeneous coordinates of $\mathbb{P}^4$. I am ...
3
votes
1answer
131 views

Decomposition of hyperbolic surfaces near cusps into annuli

Let $C=\mathbb{H}/\Gamma$ be a hyperbolic surface and $c$ a cusp of this sruface. In the paper "Billiards and Teichmüller curves on Hilbert modular surfaces" by C. McMullen, it is claimed that near ...
3
votes
2answers
162 views

Examples of toric threefolds

I am looking for examples of smooth projective toric threefolds $\mathbb P_\Delta$ such that the rational polytope $\Delta$ has only pentagonal faces and hexagonal faces. I quickly searched for ...
0
votes
1answer
179 views

Is g( ) rational if it looks that way on a large rational subset?

Let $F$ be any infinite field, $U\subset F^n$ be an open, dense (in Zariski topology) subset, $x_1,x_2,…,x_n$ be an algebraic independent system of variables over $F$ , $f,f_1,f_2,…,f_n \in ...
1
vote
1answer
307 views

Explicitly describing the region of the plane “outward of” a simple, open, oriented, cubic curve $c:(0,1)\to\mathbb{R}^2$

Some Context: I'm working with some data given in the form of Bezier curves. I need to sort these (partially ordered) Bezier curves by "outwardness" (described below) and have come across an ...
4
votes
0answers
283 views

is there a moduli of stable infinity categories?

I know there exists a groupoid-valued prestack parameterizing (connected) triangulated dg-categories (ie the points of this moduli are not objects of a fixed dg-category, but rather dg-categories ...
11
votes
1answer
337 views

K3 surface as an anticanonical section

Let $S$ be a projective K3 surface. Then is there always a smooth projective 3-fold $V$ that has $S$ as its anticanonical section?
3
votes
1answer
233 views

Castelnuovo-Mumford regularity in multigraded case

Let $R=\oplus_{n\geq 0}R_n$ be a standard Noetherian commuative graded ring over a local ring $(A,m)$ where $R_0=A.$ Put $R_+=\oplus_{n\geq 1}R_n.$ Let $M$ be a finitely generated $\mathbb Z$-graded ...
4
votes
1answer
180 views

The dualizing sheaf for a proper smooth variety

Suppose $X$ is a $n$ dimensional proper smooth variety, is the dualizing sheaf of $X$ the top wedge of sheaf of differentials: $\omega_X^0=\wedge^n\Omega^1_X$? If not what is it? (By Chow lemma, we ...
3
votes
3answers
239 views

Difference between Gieseker semistable and slope semistable

Let $X$ be a projective reduced (not necessarily irreducible) curve over an algebraically closed field and $\mathcal{F}$ be a pure coherent sheaf on $X$. Is it true that $\mathcal{F}$ is Gieseker ...
0
votes
0answers
143 views

Are del-Pezzo surfaces complete intersections?

Let $X_k$ be $\mathbb{CP}^2$ blown up at $k$-points (where $k$ is from $0$ to $8$). I think it is known that $X_k$ can be embedded in $\mathbb{CP}^n$ for some $n$. $\textbf{Question:}$ Can $X_k$ ...
7
votes
1answer
250 views

Verifying that $\epsilon^!$ is indeed the right adjoint of $\epsilon_*$ in the context of algebraic stacks

The question is about the last paragraph of Remark 12.4.3 in the book on algebraic stacks by Laumon and Moret-Bailly. Let $S$ be a (quasi-separated) scheme and let $\mathscr{X}$ be an algebraic stack ...
0
votes
0answers
94 views

Cycle class map in smooth quasi-projective varieties

Let $X$ be a smooth quasi-projective variety over $\mathbb{C}$ and $Z$ be a closed subvariety of codimension $k$. Q1. How to define a cycle class $[Z]\in H^k(X,\Omega_X^{k})$ ? Q2. More general, ...
2
votes
0answers
60 views

Geometric/algebraic interpretation of quadratic points of rank r

In the paper of Eckl/Puhklikov (http://arxiv.org/abs/1210.3715) the following terminology is introduced: " Let $X \subset Y$ be a subvariety of codimension 1 in a smooth quasiprojective complex ...
0
votes
0answers
60 views

Embedding curves in hypersurfaces

Consider a curve $C$ in $\mathbb{F}_q^m$, say. I am interested in the existence of curves not contained in any small degree hypersurface. For instance, a helix is not contained (or non-embeddable) in ...
0
votes
2answers
122 views

Is any F-stable maximal torus contained in some F-stable Borel subgroup? [closed]

Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an affine algebraic group over $\bar{\mathbb{F}}_q$, and let $F$ be a ...
2
votes
1answer
161 views

Torsion theory for quasi-coherent sheaves?

In a category $\mathcal C$, we will say that $(\mathcal T,\mathcal F)$ is a torsion theory if it satisfies: (1) $Hom(T,F)=0$ for all $T\in \mathcal T$ and $F\in \mathcal F$. (2) If $Hom(T,F)=0$ for ...
6
votes
2answers
207 views

Extending the Abel-Jacobi map over the DM-compactification $\overline{\mathcal{M}}_2$?

Let $\mathcal{M}_2$ be the moduli space of genus two curves and $\mathcal{A}_2$ the moduli space of principally polarized abelian surfaces. Then the Abel-Jacobi map gives an open embedding ...
4
votes
1answer
175 views

Macaulay's example of prime ideals in $\mathbb C[X_1,X_2,X_3]$ having large number of generators

There is a famous example of Macaulay which shows that there are prime ideals of height two in $\mathbb C[X_1,X_2,X_3]$ having at least $l$ generators for any $l\ge 3$. In Macaulay's words, the ...
3
votes
1answer
135 views

Sections of morphisms up to fppf covering

Let $f:X\to S$ be a finite type affine morphism of schemes where $S$ is an integral noetherian affine regular scheme whose function field is of characteristic zero. Assume that all geometric fibers ...
2
votes
0answers
74 views

Understanding the obstruction cone of a symmetric obstruction theory

Let $X$ be a scheme over $\mathbb C$ carrying a symmetric perfect obstruction theory $\phi:E\to L_X$, in the sense of Behrend-Fantechi (here $L_X$ is the truncated cotangent complex, in degrees ...
1
vote
0answers
147 views

Reference request: Cohomology of Elliptic Curves

Is it true that the group $$H^1(Gal(K^{ab}/K)/\mu_{\nu}(Gal(K_{\nu}^{ab}/K_{\nu})),E_{p^n})$$ is always p-divisible? Or are there any conditions which, when satisfied, guarantee its p-divisibility? ...
2
votes
1answer
267 views

An application of the Base Change Theorem to the moduli space of sheaves

Let $S$ be an abelian or K3 surface, $H$ an ample class on $S$, $v\in H^{ev}(S,\mathbb{Z})$ and $M$ the moduli space of $H$-stable sheaves on $S$ with invariants fixed by $v$. Let $\mathcal{E}$ be a ...
0
votes
0answers
83 views

Surjectivity of ring homomophism induced by Frobenius endomorphism

Denote by $F_q$ the finite field with $q$ elements, and denote by $\bar{F_q}$ its algebraic closure. Let $V$ be an affine $\bar{F_q}$-variety and $F$ be the Frobenius endomorphism corresponding to an ...
2
votes
1answer
96 views

Can height one maximal ideals in the normalization contract to non-height one primes in the base?

Let $R$ be a local (Noetherian) integral domain of dimension greater than one. Can the integral closure (i.e. normalization) of $R$ have a maximal ideal of height one?
2
votes
1answer
106 views

Classification of local and semi-local rings in function fields

Let $C$ be a non-singular algebraic curve over an algebraically closed field $k$, and $F$ a function field of this curve. It is well-known that non-trivial discrete valuation rings of $F$ correspond ...
4
votes
1answer
122 views

Hilbert point and Hilbert stability

For $X\in \mathbb{P}^N$ a closed subscheme, one can consider the m-th Hilbert point $$ [X]_m=[\bigwedge^{h^0(X, \mathcal{O}(m))}H^0(\mathbb{P}^N, \mathcal{O}(m))\to \bigwedge^{h^0(X, ...
3
votes
2answers
213 views

Upper bound on Betti numbers of an intersection of hypersurfaces (or quadrics)

I have a problem I have been stuck with since several weeks now, and yet I believe it should be easy to specialists. Let $k$ be an algebraically closed field, $m$ and $n$ two integers. Let ...
0
votes
0answers
93 views

How to show integrally closed implies topologically unibranch

On p.52 of Mumford's book Algebraic Geometry: Complex projective varieties, he states that $$\mathcal{O}_{x.X} \text{is integrally closed} \ \Rightarrow X \ \text{is topologically unibranch at } \ ...
8
votes
1answer
330 views

Group schemes, adeles, double cosets, and étale cohomology

Let $K$ be a number field, $R$ the ring of integers of $K$, ${\mathbf{A}^f}$ the ring finite adeles of $K$, and ${\widehat{R}}\subset {\mathbf{A}^f}$ the ring of integral adeles. Let $G$ be an affine ...
6
votes
4answers
652 views

Is there a (satisfying) proof that cellular cohomology is isomorphic to simplicial cohomology that doesn't use relative cohomlogy?

That singular and de Rham cohomologies of a smooth manifold are isomorphic has two proofs that I know of. The classical one uses Stokes' theorem to give the isomorphism explicitly. The second proof ...
3
votes
1answer
114 views

I need to refind a reference on multigraded Hilbert series

I found a theorem about multigraded Hilbert series stated as follows: Let $R$ be a Noetherian multigraded algebra $R:=\bigoplus_{j\in\mathbb{N}^m}{R_j}$ over $R_0=\mathbb{C}$. If $R$ is generated by ...
2
votes
0answers
83 views

Extension of the Hilbert-Mumford Criterion

Let $X$ be a smooth variety, $L$ a line bundle on $X$ and $G$ a reductive group actin on $X$ with a linearization of the action to $L$. Say we are over the complex numbers. Both the concept of GIT ...
3
votes
0answers
72 views

Low height integer points on a rank variety

Let $M_i$ be fixed rectangular matrices with integer coefficients less than $n$. Consider the variety defined by the condition $$ \mathrm{rank}(\lambda_1M_1 + \lambda_2M_2 + ... + \lambda_kM_k) = 1. ...
5
votes
0answers
100 views

Injective homomorphism induced by cup product in cohomology

Let $M$ be an irreducible holomorphic symplectic manifold of dimension $\geq 4$. In his paper 'A survey of Torelli and Monodromy results', Markman claims (discussion after Theorem 9.7) that the cup ...
3
votes
1answer
122 views

Graded ring of a genus 2 curve

Let $X$ be a smooth projective complex curve of genus 2 with canonical divisor $K$. $X$ of course is hyperelliptic and has an involution that I denote by $j$. There exists 3 possibilities for ...
-3
votes
1answer
177 views

Loop space of manifold [closed]

Question A: The free loop space of a manifold is also a manifold? Question B: The free loop space of an algebraic variety is also a algebraic variety ? Are these questions asked or answered anywhere ...
5
votes
0answers
128 views

Line-configuration in algebraic hypersurfaces

Suppose we are working on $\mathbb{C}^3$ (so that maybe this is more appealing to algebraic geometers) and we are given an algebraic equation $F(x,y,z)=0$ of degree $n\geq 3$ defining some smooth ...
1
vote
2answers
134 views

Canonical (tautological) section of a family of sheaves

A couple of months ago, i saw a construction, that somehow looks like the construction of the tautological section of the pullback of a vector bundle to its total space, i am trying to piece it ...