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

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4
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1answer
174 views

Can every genus $2$ curve be written as ramified cover of elliptic curve?

Suppose $C$ is a curve of genus $2$, does $C$ admit a surjective morphism onto some elliptic curve $E$?
9
votes
1answer
306 views

Embedding linear algebraic groups of a given dimension into a fixed $\mathrm{GL}_N$

Given $n$, can $n$-dimensional linear algebraic groups over $\mathbb{C}$ be embedded into $\mathrm{GL}(N,\mathbb{C})$ for a uniformly bounded $N$? Thanks so much for your reply!
1
vote
1answer
193 views

A $d$-form on ${\mathbb R}^n$ that vanishes on $\binom{d+n-1}{n-1}$ general points, vanishes identically

I'm looking for a reference for the fact that a $d$-form on ${\mathbb R}^n$ that vanishes on $p_1,..,p_{\binom{d+n-1}{n-1}}$ general points, vanishes identically. A specific construction of a set of ...
0
votes
0answers
39 views

center of divisorial valuations on toric varieties

Suppose we are given a divisorial valuation $\mathcal{v}$ on a smooth toric variety $X_{\Sigma}$ (for a fan $\Sigma$), i.e. $\mathcal{v}$ is the valuation induced by a torus invariant divisor ...
1
vote
0answers
97 views

A question on exponential sums

Let $F(x_1, \cdots, x_n) \in \mathbb{Z}[x_1, \cdots, x_n]$ be a homogeneous polynomial, irreducible over $\mathbb{Q}$. Let $A$ be a positive constant, and let $B$ be a positive real number understood ...
0
votes
0answers
69 views

Finding a set of generators of an ideal with certain property in $k[x_1, …, x_n]$

I am interested in the following problem, and I would appreciate any comments, inputs, answers, references! Let $k$ be a field. For each $1 \leq j \leq n$, let $$ I_j = (x_j, u_{2}^{(j)}, ..., ...
1
vote
0answers
90 views

cohomlogy of Diagonal ring

Let $S=\bigoplus_{\underline n\in\mathbb N^r } S_{\underline n}$ be a standard multigraded ring over a local ring and M be a finitely generated $\mathbb N^r $-graded $S$-module. Let ...
3
votes
0answers
152 views

Kähler differentials, intuition behind $\text{div}(\omega)$, canonical divisor on algebraic curves?

See my two previous questions here: Intuition for thinking about R-module of Kähler differentials, universal receptacles, derivations? and Kähler differentials, define valuation? for background. If ...
-1
votes
0answers
68 views

Canonical divisor of a Fano cyclic cover

Let $\pi: Y\to X$ be the $m$-cyclic cover of smooth varieties branched along a smooth divisor $D$ and determined by $\mathcal{L}$, where $\mathcal{L}^{\otimes m}=\mathcal{O}_X(D)$. Then ...
11
votes
0answers
182 views

q-Catalan numbers from Grassmannians

In this question by $q$-Catalan numbers I mean the $q$-analog given by the formula $\frac{1}{[n+1]_q}\left[{2n\atop n}\right]_q$. The polynomial $\left[{2n\atop n}\right]_q$ represents the class of ...
0
votes
0answers
100 views

Free resolutions of affine (non-projective!) varieties

Say, you have an ideal $I$ of a polynomial ring $R = K\lbrack X_1,\ldots,X_n \rbrack$ over an algebraically closed field $K$ (you can assume $K = \mathbb{C}$). What does a minimal free resolution of ...
1
vote
1answer
133 views

Completion of discrete valuation ring

Let $R$ be an excellent, Henselian, discrete valuation ring with algebraically closed residue field and $\hat{R}$ be the completion of $R$. If I understand correctly, the residue field of $\hat{R}$ is ...
0
votes
1answer
122 views

Change of grading used in the paper “The diagonal subring and the Cohen-Macaulay property of a multigraded ring” by Eero Hyry

I am currently reading the paper "The diagonal subring and the Cohen-Macaulay property of a multigraded ring" by Eero Hyry. I do not understand the following part in Lemma 1.1. here. Let ...
-1
votes
0answers
88 views

Sheaf of rank 1 on smooth projective variety

Let $X$ be a smooth projective variety over $\mathbb{C}$ and let $F$ be a torsion free sheaf of rank $1$ on $X$. Then why $F\cong I\otimes L$ (in a unique way), where $I$ an ideal sheaf of finite ...
2
votes
0answers
98 views

Kähler differentials, define valuation? [migrated]

See my previous question for a definition of the $K$-module of Kähler differential $\Omega_{K/k}$. This question is sort of a follow up on it. Suppose $k$ is a field of characteristic $0$, $R$ is a ...
6
votes
0answers
93 views

Geometric interpretation of Schmidt rank

For a form $f \in k[x_1, \cdots, x_n]$, where $k$ is a field of characteristic zero (or more specifically, a number field, and usually $\mathbb{Q}$). The Schmidt rank of $f$ (with respect to $k$), ...
6
votes
1answer
172 views

Pontryagin Forms and Special Holonomy

Let $(M,g)$ be a Riemannian manifold. Recall that the $k^{th}$-Pontryagin class is a topological invariant which, by classical Chern-Weil theory, can be represented using the so-called Pontryagin ...
0
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0answers
122 views

Determining a scheme $X$ is affine from $Qcoh(X)$

My question is a subquestion of this question. And a repost from this MSE question. The settings is the following. Let $X$ be a scheme. Assume that the adjunction in TAG01BH of the stacks project is ...
5
votes
1answer
175 views

Intuition for thinking about $R$-module of Kähler differentials, universal receptacles, derivations?

Suppose $k$ is a field of characteristic zero, and $R$ is a $k$-algebra. The $R$-module of Kähler differentials $\Omega_{R/k}$ of $R$ over $k$ with generators $\{dr\}_{r \in R}$ is the module subject ...
13
votes
0answers
299 views

Is there a motivic Cauchy integral formula?

Let $R$ be a complete dvr with fraction field $K$ and residue field $k$, and let $X, Y$ be two smooth projective $R$-schemes with isomorphic generic fibers. Is it true that $[X_k]=[Y_k]$ in ...
0
votes
1answer
123 views

Chow groups of rational varieties

Let $X$ be a smooth projective rational variety over a field $k$. Let $CH^i(X)$ denote the Chow group of codimension $i$ algebraic cycles on $X$ modulo rational equivalence. What can one say about ...
4
votes
1answer
206 views

Picard groups of Fano varieties in positive characteristic

Let $k$ be an algebraically closed field of characteristic $p \geq 0$. Let $X$ be a smooth Fano variety over $k$ and let $\ell \neq p$ be a prime. Is the natural morphism $\mathrm{Pic}(X) \otimes ...
0
votes
0answers
91 views

Classification of compact Shimura curves

Is there a classification that determines all isomorphism classes of compact Shimura curves at least Shimura curves in $A_g$? I did not find this in the literature and appreciate any helpful ...
7
votes
1answer
104 views

Elementary proof of a triangular grid lemma

I am looking for an elementary proof of the following lemma, which concerns what Green and Tao call "triangular grids" (see arXiv:1208.4714). Let $a_1$, $a_2$, $a_3$, $a_4$, $b_1$, $b_2$ be six ...
1
vote
1answer
208 views

Strong form of Grothendieck's algebrization theorem

Let $R$ be a Henselian discrete valuation ring with algebraically closed residue field $k$ ($R$ is not necessarily complete), $X$ a regular surface over $\mathrm{Spec}(R)$ and a sequence of locally ...
1
vote
1answer
139 views

What is the moduli functor $\mathbb{P}Ext^1(L,M)$ represents

Let $X$ be an algebraic space and $L,M$ are vector bundles with rank $n,m$. Then, It is known that $\mathbb{P}Ext^1(L,M)$ is a parameter space for isomorphism classes of vector bundle of rank $n+m$, ...
0
votes
1answer
104 views

polynomial expression for counting number of integral points of a set

Let $v_i=a_ie_i\in\mathbb R^d$ and $w_i=b_ie_i\in\mathbb R^d$ for $i=1,\dots,d$ where $e_i$'s are unit vetcors and $a_i,b_i$ are positive integers. Let $$S=conv\{0,rv_i+sw_i:i=1,\dots,d\}.$$ Can we ...
1
vote
0answers
69 views

A question about extension of relative subvariety

Let $X\rightarrow S$ be a holomorphic family of projective manifolds. My question is: can we find a Zariski open set $S'\subset S$ such that for any $t\in S'$, any given subvariety $Y\subset X_t$ with ...
0
votes
0answers
100 views

Geometric transitions of Calabi-Yau threefolds

Let me start with the definition of geometric transition: Let $Y$ be a Calabi–Yau 3–fold and $\phi: Y \rightarrow \overline Y$ be a birational contraction onto a normal variety. If there exists a ...
-6
votes
1answer
221 views

Quintic Equation [closed]

Can we solve the following polynomial quintic equation by radicals x^5 + x^4 = 1 I found one real root which is algebraic solution (no approximation method ...
1
vote
1answer
166 views

A key step in Del Busto's effective Matsusaka theorem

I'm reading a proof of del Busto (http://arxiv.org/pdf/alg-geom/9410018v1.pdf) and am confused at one step. I'm sure its quite easy, but the key trick hasn't come to me yet. The context here is that ...
2
votes
0answers
93 views

relations in (\mathbb P^1)^n

What is a minimal set of relations of the image of $(\mathbb P^1)^n$ in the Segre embedding? For $n=2$ its just the determinant $x_1x_2.y_1y_2=x_1y_2.x_2y_1$. Is it written explicitly somewhere?
1
vote
1answer
139 views

On functors which are generically representable

Let $F$ be a set-valued (contravariant) functor on the category of schemes. Let $F_{\mathbb Q}$ be the associated functor on the category of schemes over $\mathbb Q$. Suppose that $F_{\mathbb Q}$ is ...
2
votes
0answers
63 views

Question related to $h$-invariant of a form

Let $k$ be a field. Given a form $f \in k[x_1, ..., x_n]$ of degree at least $2$, we define the Schmidt rank, also known as the $h$-invariant, $h_k(f)$ to be the least positive integer $h$ such that ...
3
votes
0answers
149 views

Is there a Hochschild-Serre spectral sequence for unramified cohomology?

Similar to the Hochschild-Serre spectral sequence for etale cohomology ($H^p(G, H^q_{et}(X_L, \mathcal F|_{X_L})) \Rightarrow H^{p+q}_{et}(X, \mathcal F)$ for a Galois field extension $L/k$ with ...
2
votes
1answer
138 views

Constant spinors from constant forms

Let $(X,g)$ be a $m$-dimensional complex, hermitian, spin manifold and let us denote by $S_{\mathbb{C}}$ its complex spinor bundle. Then: $S_{\mathbb{C}}\simeq \Lambda_{\mathbb{C}}(X)$ Let $\nabla$ ...
1
vote
0answers
55 views

About properties of polynomials with common interlacing

Say $\{a_1,a_2,..,a_n\}$ and $\{b_1,b_2,...,b_n\}$ be the real roots of two monic polynomials of degree $n$ which have a common interlacing. (say I have arranged the roots in increasing order) Can ...
9
votes
1answer
325 views

Excellent rings

If A is an excellent commutative ring and G is a finite group of automorphisms of A, is the invariant subring A^G still excellent ? I think this is false -- because if not it would probably be written ...
2
votes
0answers
100 views

The existence of proper schemes under complection

Let $R$ be a regular local ring, $\hat{R}$ be its completion, $X$ be a proper scheme over $\text{Spec}(\hat{R})$. In what case there exist a proper scheme $Y$ over $\text{Spec}(R)$, such that $X$ is ...
6
votes
1answer
155 views

Question about zeta function of function field in 1 variable over $\mathbb{F}_q$

From my previous question, I know that$$\zeta_X(s) = {{P(u)}\over{(1-u)(1-qu)}}$$for some polynomial $P(u)$ of degree $2g$, where $X$ is the set of all places of $F$, a function field in one ...
2
votes
0answers
69 views

Rational curves through a fixed number of points

Let us fix two positive integers $d$, and $N$. Can we determine a third integer $n$ such that given $n$ general points $p_1,...,p_n\in\mathbb{P}^N$ there exists a unique rational curve of degree $d$ ...
7
votes
1answer
255 views

Reference request, zeta function is rational function via Riemann-Roch?

I am looking for a reference to a proof that the zeta function of a function field in one variable over a finite field $\mathbb{F}_q$ is a rational function in $q^{-s}$ by using the Riemann-Roch ...
0
votes
1answer
143 views

Moment maps and flat degenerations of toric varieties

We have a flat family of projective varieties with a torus $T$ action, over $\mathbb{A}^1$. How do the moment map images of the fibers change when we pass from the generic fiber to the special fiber ...
0
votes
0answers
43 views

Picard sequence for sujective morphisms

Given $\phi:X\rightarrow Y$ a surjective morphism of $k$-algebraic varieties ($k$ separably closed), I wanted to find how the write an exact sequence involving Pic(X) and Pic(Y). We can use the long ...
2
votes
1answer
197 views

Rational curves in projective spaces

Let $X\subset(\mathbb{P}^{N})^n$ be the variety defined as follows: $(p_1,...,p_n)\in (\mathbb{P}^{N})^n$ such that there exists a rational curve $C$ of degree $d$ with $p_1,...,p_n\in C$. Is there a ...
2
votes
0answers
62 views

Generalized Hurwitz Spaces

In this question all the varieties are over $\mathbb{C}$. Classic Hurwitz spaces $\mathcal{H}_{g,r}$ are moduli spaces of simple branched coverings $f \colon X \to \mathbb{P}^1$ of degree $d$, where ...
1
vote
1answer
110 views

canonical divisors of a resolution of a normal surface singularity

Let $(0\in X)$ be the germ of a normal surface singularity and let $f: Y \to X$ be the minimal resolution. Questions> (1) How can I define a map $f_*\mathcal{O}_Y(K_Y)\hookrightarrow ...
5
votes
1answer
191 views

Are genus zero Gromov Witten Invariants on Del-Pezzo surfaces enumerative?

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 the homology class given by $$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$ ...
5
votes
0answers
184 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
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0answers
139 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 ...