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

**5**

votes

**0**answers

189 views

### Hyperplane sections of principal homogeneous spaces

Let $P_i$ denote the $i$-th vertex in the Dynkin diagramm of an algebraic group. It symbolizes a parabolic subgroup of $G$ corresponding to the other vertices, meaning $G/P_i$ is a smooth, projective, ...

**0**

votes

**0**answers

141 views

### Soft Question: Relationships Between Moduli Space and Objects They Parametrize

Apologies in advance if this question is not suitable for MO. My friend and I were wondering recently what, if any, are the relationships between the geometric properties of a moduli space and the ...

**0**

votes

**0**answers

40 views

### Singularities of the product of a $(\mathbb{C}^*)$-surface with $\mathbb{C}$

Recall that any normal $\mathbb{C}^*$-surface is Cohen-Macaulay
and there exists normal $\mathbb{C}^*$-surfaces whose
singularities are not rational.
Does anyone know an example of a normal ...

**1**

vote

**1**answer

103 views

### Schubert Polynomials for Complex Projective Space

The Borel picture of the cohomology ring of a flag variety gives a description as a coset space, without identifying any representatives for the classes. Lascoux and Schützenberger gave specific ...

**1**

vote

**0**answers

80 views

### On the compactification of moduli space of vector bundles

Let $X$ be an irreducible, nodal curve over an algebraically closed field of genus at least $2$. Denote by $U(r,d)$ (resp. $U^0(r,d)$) the moduli space of torsion-free (resp. locally free) sheaves of ...

**3**

votes

**2**answers

138 views

### Obstruction to get a galois invariant cycle

Let $X$ be a smooth projective variety over a finite field $k$, $G=Gal(\bar{k}/k)$ and $\Gamma\in CH^i(\bar{X})$ such that:
$cl(\Gamma) \in H_{et}^{2i}(\bar{X},\mathbb{Z}_l(i))^G$ and
$\exists$ ...

**4**

votes

**0**answers

112 views

### Reference for Grothendieck's duality and Cousin, Dualizing and Residual complexes

I am a graduate student currently reading Hartshorne's Residues and Duality. In order to reach the construction of the right adjoint $f^!$ of $Rf_*$ for some special types of maps of locally ...

**2**

votes

**0**answers

172 views

### On Abelian Galois Covering

Consider a complete quadrangle $\Delta$ in $\mathbb{CP}^2$ (i.e. the union of the six
lines through points $P_1$, $P_2$, $P_3$ and $P_4$ in general position). Let $f: Y := \hat{\mathbb{CP}^2}(P_1, ...

**3**

votes

**1**answer

203 views

### What are the indecomposable classes on a del-Pezzo surface?

Let $X_k$ be $\mathbb{P}^2$ blown up at $k$ points (where $k$ is $0$ to $8$).
Let $\beta \in H_2(X_k, \mathbb{Z}) $ be a homology class given by
$$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$
...

**3**

votes

**0**answers

112 views

### Does there exist a continuous surjection? [closed]

Let $C$ be an irreducible projective cubic in $\mathbb{P}_2$ with a singular point $p$. So consider $f: \mathbb{P}_1 \to C$ defined as follows. Identify $\mathbb{P}_1$ with the set of lines in ...

**3**

votes

**0**answers

124 views

### Varieties acted upon faithfully by an abelian variety

Let $X$ be a smooth projective variety over the complex numbers. Suppose that some positive-dimensional abelian variety $A$ acts faithfully on $X$.
Examples of such varieties $X$ are provided by ...

**3**

votes

**1**answer

262 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

**0**answers

257 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

**1**answer

162 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

**1**answer

141 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

**0**answers

155 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

**0**answers

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

**1**answer

122 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

**1**answer

188 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

**2**answers

200 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

**2**answers

193 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 ...

**9**

votes

**1**answer

518 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

**2**answers

185 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

**1**answer

134 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

**2**answers

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

**1**answer

181 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 ...

**2**

votes

**1**answer

342 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

**0**answers

287 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

**1**answer

340 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

**1**answer

235 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

**1**answer

181 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

**3**answers

247 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

**0**answers

149 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 $X_k$ ...

**10**

votes

**1**answer

312 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

**0**answers

98 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

**0**answers

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

**0**answers

62 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

**2**answers

127 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

**1**answer

164 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

**2**answers

212 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

**1**answer

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

**1**answer

136 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 ...

**3**

votes

**1**answer

110 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 ...

**2**

votes

**0**answers

151 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

**1**answer

270 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

**0**answers

86 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

**1**answer

97 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

**1**answer

107 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

**1**answer

126 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

**2**answers

219 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 ...