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

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votes

**0**answers

182 views

### Unirationality over $\mathbb{Q}$

It is known that all smooth projective quartic hypersurfaces of suitably large dimension are unirational over $\overline{\mathbb{Q}}$. Are there any results regarding unirationality over $\mathbb{Q}$ ...

**14**

votes

**3**answers

1k views

### Euler characteristic of a curve

Is it a complete coincidence that the Euler characteristic of a curve is exactly twice the Euler characteristic of its structure sheaf?

**2**

votes

**0**answers

183 views

### Is there a well-known tautological bundle over $\mathbb{P}(\Lambda^nV)\times V$?

Let $V$ be a vector space of dimension $>n$, and define the subset $$
K:=\{ ([\omega],v)\mid v\wedge\omega=0 \}\subset\mathbb{P}(\Lambda^nV)\times V\, .
$$
Denote also by $\pi:K\longrightarrow ...

**0**

votes

**1**answer

222 views

### Etale cohomology and topological invariance

Let $X$ be a projective scheme and $X_0 \subset X$ a subscheme defined by a nilpotent ideal. Denote by $i:X_0 \to X$ the closed immersion. Let $\mathcal{F}$ be a locally free sheaf sheaf on ...

**11**

votes

**2**answers

456 views

### Is there a “free abelian group of rank 1” in the category of affine group schemes?

Let's fix an algebraically closed field $k$.
The group $\mathbb Z$, as a discrete group scheme, is not affine since it's not quasi-compact. Is there an affine algebraic scheme over $k$ whose ...

**1**

vote

**0**answers

146 views

### Separability and smoothness

Let $A \subseteq B$ be commutative noetherian rings.
I have found the following claim: "Separability implies smoothness" with the following explanation:
"The natural thing is to prove that a separable ...

**3**

votes

**1**answer

145 views

### Example of a homogeneous (not monomial) $(x,y)$-primary ideal $I$ in $K[x,y]$

Is there any example of a homogeneous (not monomial) $(x,y)$-primary ideal $I$ in $K[x,y]$ such that $I$ is complete and and there exists a minimal reduction $J$ of $I$ such that ...

**0**

votes

**0**answers

97 views

### Zariski closure construction (over a field) implicitly uses reduced induced structure?

Let $X$ be a $k$-scheme of finite type, $k$ a field, and $\Sigma$ a subset of $X(k)$. Since the $k$-points of $X(k)$ may be identified with the physical points $x\in X$ for which the structural map ...

**4**

votes

**0**answers

123 views

### Is there a concrete description of $\Theta_{sing}$ for a generic curve of genus 6?

If C is generic of genus 6, then $\Theta_{sing}$ is a smooth surface. Can anyone give me a reference or a hint as to what that surface might be. What are the numerical characteristics of this surface? ...

**12**

votes

**2**answers

449 views

### Infinitely many irreducible polynomials of the form f(X^2) + X mod 3?

Are there infinitely many polynomials $f \in \mathbb{F}_3[X]$ for which $f(X^2) + X$ is irreducible?

**5**

votes

**0**answers

187 views

### Genus of $k(T)$ is $0$ without using Riemann-Roch

Let $F$ be a function field with one variable with total constant field $k$, and let $X$ be the set of all places of $F$. How do I show that the genus of $k(T)$ is $0$ without using Riemann-Roch? Is ...

**1**

vote

**1**answer

204 views

### Two limit cycles which lie on the same leaf

Edit 1: For a related discussion see this MSE post
I apologize in advance, if this question is obvious:
1)What is an example of a polynomial vector field on $\mathbb{R}^{2}$ with at least two ...

**0**

votes

**1**answer

104 views

### Complex plane mod lattice to elliptic curve correspondence generalization

If we observe the correspondence
$$\mathbb{C}/\Lambda \rightarrow E: Y^{2} = X^{3} - \frac{g_{2}(\Lambda)}{4}X - \frac{g_{3}(\Lambda)}{4},$$
we see the relationship between weight 4 and weight 6 ...

**2**

votes

**1**answer

119 views

### Relative dualizing sheaf (reference, behavior)

Let $\mathcal{C}\rightarrow S$ a flat projective family of locally complete intersection projective curves over a integral noetherian scheme (say a spectrum of a local ring). I was wondering whether ...

**1**

vote

**0**answers

48 views

### Intersection of two components and Jacobian criterion

A reducible scheme is singular along the intersection of two components: this is usually proved using the well-known, but not entirely trivial, fact that a regular local ring is an integral domain. ...

**2**

votes

**0**answers

65 views

### Non canonical singularities of moduli spaces of curves

Is it true that for any $g\geq 1$ and $n$ such that $\overline{M}_{g,n}$ has dimension at least two the locus in $\overline{M}_{g,n}$ parametrizing reducible curves which are union of an elliptic ...

**1**

vote

**0**answers

63 views

### Gluing subschemes of fibers of Hilbert Scheme (in mixed characteristic)

Let $\mathfrak{X}\rightarrow Spec(R)$ be a smooth family of smooth projective varieties over a local 1-dimensional ring of mixed characteristic. Suppose that there are non-empty subschemes (locally ...

**6**

votes

**1**answer

236 views

### Is there a tropical geometric proof for counting genus g curves in any n dimensional projective space?

Consider the following question: Let $X$ be a compact complex manifold
and $\beta \in H_2(X, \mathbb{Z})$ a fixed homology class. Let
$\mu_1, \mu_2, \ldots, \mu_k$ denote certain generic ...

**12**

votes

**1**answer

421 views

### Which nice/deep elaborations on the (operators <-> sheaves) / (endomorphisms <-> objects) theme are there?

A linear operator $T:V\to V$ on a (say) vector space over a field $k$ is just a $k[T]$-module, and may be viewed as the sheaf $\mathscr F_T$ over $\mathbb A^1_k$, with fibre over $\lambda\in k$ equal ...

**1**

vote

**0**answers

79 views

### irreducibility of varieties defined by square roots equatiosn

I have the following collection of equations in matrix form:
\begin{equation}
\sqrt{\pi_{ij}\pi_{ik}\pi_{jl}\pi_{kl}}+ \sqrt{\pi_{ik}\pi_{il}\pi_{jk}\pi_{jl}}+\sqrt{\pi_{il}\pi_{ij}\pi_{kl}\pi_{jk}} ...

**3**

votes

**0**answers

65 views

### $H^{1}(C, N_{C/X}(-m)) = 0$, for $C$ a irreducible curve on $X$ through $m$ general points

I was reading A. de Jong and J. Starr's paper "Low degree complete intersections are rationally simply connected", which can be found at http://www.math.stonybrook.edu/~jstarr/papers/nk1006g.pdf, and ...

**4**

votes

**1**answer

163 views

### Infinitesimal deformations of fake projective planes (or ball quotients)

This question is related to the answer I gave to this MO question. What I'm asking is probably well-known to the experts in the field, and I apologize in advance if this turns out to be trivial.
By ...

**6**

votes

**1**answer

386 views

### Analytic Chern classes

I have two questions on Chern classes, following Huybrechts' Complex Geometry.
Are the analytic Chern forms just the elementary symmetric polynomials of the eigenvalues of the curvature?
I googled ...

**4**

votes

**2**answers

302 views

### Infinitesimal deformations of a fibration

Let $f:X\rightarrow Y$ be a morphism of normal projective varieties over an algebraically closed field with connected fibers.
Assume that both $Y$ and the general fiber of $f$ admit a non-trivial ...

**-3**

votes

**0**answers

98 views

### Non-flat $R \subseteq S$, which is integral, separable, $R$ is a noetherian (not integrally closed) integral domain

On ramification theory in noetherian rings,
of Auslander and Buchsbaum say:
"Chapter 4 is devoted to showing that under various conditions if $S$ is unramified over $R$, then $S$ is $R$-projective. ...

**0**

votes

**0**answers

71 views

### higher direct images of relative canonical sheaf plus a fractional divisor

For a map $f: Y \rightarrow X$ branched over simple normal crossing divisor $B=\sum_iB_i$, do we know of similar local freeness property for higher direct image of relative canonical sheaf plus a ...

**1**

vote

**0**answers

112 views

### Riemann-Roch formula for nodal curves

Let $X$ be an irreducible, reduced, projective curve over an algebraically closed field, with at worst nodes as singularities. Let $\mathcal{F}$ be a trivial vector bundle on $X$ of rank $r$. Consider ...

**7**

votes

**2**answers

451 views

### What is descent data (of higher categories), conceptually?

First consider a scheme $X$ with an open cover $\mathcal{U}=\{U_i\}$. An object with descent data on $\mathcal{U}$ is a collection $(\mathcal{E}_i,\phi_{ij})$ where $\mathcal{E}_i$ is a ...

**6**

votes

**2**answers

854 views

### About Abhyankar's conjecture

I just came to this conjecture (proved by M.Raynaud and D.Harbater in 1994) last weekend, in Fresnel and v.d.Put's book Rigid Geometry and Its Applications. It claims that all quasi $p$-group $G$ ...

**21**

votes

**2**answers

409 views

### Equivalence of “Weyl Algebra” and “Crystalline” definitions of rings of differential operators between modules?

Let $B$ be a commutative $A$-algebra, and let $M$, $N$ be two $B$-modules. We can talk about the set of $A$-linear module homomorphisms $M \to N$, i.e. the set $\text{Hom}_A(M, N)$. Differential ...

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votes

**0**answers

93 views

### Two questions about sections of ruled surfaces

The following questions maybe elementary, but I can't find them in the literature.
Assume now everything I will write is defined over some algebraically closed field. Let $S$ be a (geometrically) ...

**14**

votes

**1**answer

369 views

### Geometric generic fibre

This is a pretty elementary question about schemes, but it came up in the course of research, so let's try it here rather than MSE.
Question 1: Are the fibres of a family of complex varieties ...

**0**

votes

**1**answer

152 views

### Is the Gysin morphism equivariant?

Let $X$ be a smooth, projective complex variety and $j \colon D \hookrightarrow X$ a smooth divisor. Then we have a Gysin morphism in singular cohomology
$$
j_\ast \colon H^{\bullet}(D) \to ...

**-1**

votes

**0**answers

103 views

### picard group of a cyclic cover

I want to find any result about the picard group of a ramified double cover of a projective plane.
Isn't there any general result about this case unlike ruled surface?
Can you recommend any good ...

**5**

votes

**0**answers

182 views

### Log smooth models for abelian varieties

Let $K$ be a field which is complete for a discrete valuation. Assume that the residue field has characteristic $p > 0$. Let $A$ be an abelian variety over $K$ having the property that (for some ...

**-1**

votes

**0**answers

101 views

### Bijection between dominant rational maps and morphisms of function fields? [migrated]

Let $X$ and $Y$ be two integral schemes of finite type over a field $k$. Consider the function fields $K(X)$ and $K(Y)$.
Do we have a bijection between:
(a) Dominant rational maps $X \rightarrow Y$
...

**2**

votes

**0**answers

97 views

### Twisting by a multiplicative Character in Katz, Perversity and Exponential sums

Let $C(x_1,\ldots,x_n)$ be a nonsigular cubic form with integral coefficients.
In his Proof that $C$ fulfills the Hasse-Principle, if $n\geq 9$, Hooley used the following estimate that was provided ...

**2**

votes

**0**answers

59 views

### TTF triples are recollements

The notion of recollement
$$
\mathcal{A}'
\stackrel{\overset{i^*}{\longleftarrow}}{\stackrel{\overset{i_*}{\longrightarrow}}{\underset{i^!}{\longleftarrow}}}
...

**1**

vote

**0**answers

79 views

### $n$-recollements and perverse t-structures

A recent preprint on arXiv brought my attention on the notion of $n$-recollement (def. 2) a generalization of the notion of recollement among three abelian or triangulated categories behaving like a ...

**1**

vote

**0**answers

58 views

### Beauville's Integrable System with singular spectral curves

Let us consider Beauville's Integrable System. So, we live on $\mathbb{P}^1$. There is the moduli space of matrices $M_r(d)/\mathrm{PGL}(r)$ with polynomial entries of degree less than or equal $d$. ...

**1**

vote

**0**answers

78 views

### All relations among degree n monomials in n variables

In the course of my work, I have run into the problem of finding exactly all relations among degree $n$ monomials in $k[x_1,\dotsc,x_n]$, with specific interest in the case $n=3$ (e.g. $x_1^2 x_2 ...

**5**

votes

**2**answers

728 views

### Idea of using etale site

I have just read an article which mentions that, when Grothendieck considered using etale morphism, he did borrow the idea from Riemann that multivalued function on an open subset of complex plane ...

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votes

**0**answers

54 views

### Proof of formula for dimension of moduli of stable vector bundles on smooth curves [migrated]

Let $C$ be a smooth curve of genus $g \ge 2$ over an algebraically closed field of positive characteristic. If I understand correctly, the dimension of the moduli space of vector bundles on $C$ of ...

**0**

votes

**1**answer

140 views

### Model over DVR for smooth projective curves

Let $C$ be a smooth, projective, geometrically irreducible curve of genus at least $2$ over a complete discrete valued field $F$ of characteristic zero (not necessarily algebraically closed). Let $R$ ...

**0**

votes

**0**answers

102 views

### residue formula for connections on curves

Let $X$ be a smooth, projective curve over a field $k$ (characteristic zero is enough for me) and $E$ a line bundle on $X$. Assume that $E$ is equipped with an integrable logarithmic connection ...

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votes

**0**answers

108 views

### Transversal intersection in the moving lemma

Let $X$ be a smooth projective variety over an algebraically closed field and let $A,B$ be closed irreducible subvarieties of $X$. Chow's moving lemma which is proved in the book by Eisenbud and ...

**2**

votes

**2**answers

224 views

### Vanishing of higher direct image of a morphism with generic fiber $\Bbb{P}^1$

The following question was asked on math.stackexchange.com with no reply for the past week or so. Let $f : X \to Y$ be a morphism of smooth (integral) varieties over $\Bbb{C}$ with generic fiber equal ...

**1**

vote

**1**answer

126 views

### The canonical bundle of an infinitesimal deformation

Let $X_0$ be a smooth projective variety over the complex numbers and let $X$ be an infinitesimal deformation of $X_0$ over the ring of dual numbers.
If the canonical bundle of $X_0$ is ample (resp. ...

**2**

votes

**1**answer

136 views

### isogeny clases of CM abelian varieties

Let $A$ be an abelian variety defined over $\overline{\mathbb{Q}}$ and with complex multiplication by a CM field $K$. Looking at the action of $K$ on $H^0(A, \Omega^1_A)$ one gets a CM type of $K$, ...

**0**

votes

**1**answer

149 views

### Smooth morphism to homogeneous spaces and fibers

Let $f:X \to Y$ be a smooth morphism between projective varieties. Suppose $Y$ is a homogeneous space. Under what additional condition on $f$, can we conclude that every fibers of $f$ are isomorphic?