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

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2
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0answers
74 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
0answers
45 views

TTF triples are recollements

The notion of recollement $$ \mathcal{A}' \stackrel{\overset{i^*}{\longleftarrow}}{\stackrel{\overset{i_*}{\longrightarrow}}{\underset{i^!}{\longleftarrow}}} ...
1
vote
0answers
64 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
0answers
44 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
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0answers
60 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 ...
4
votes
1answer
359 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 ...
0
votes
0answers
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
1answer
125 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 $R$ ...
2
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0answers
47 views

Relation between linear independence of lattice vectors and the toric variety defined by that lattice

I have been reading some basic and elementary work on toric varieties, but even though people assured me that toric varieties are very well understood, several questions remained. Setup. Let ...
0
votes
0answers
98 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 ...
0
votes
0answers
98 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 ...
1
vote
2answers
174 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
1answer
112 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
1answer
120 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
1answer
136 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?
1
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0answers
95 views

Artin's criterion for étale, quasi-separated algebraic spaces

it is known from Knutson's work that an algebraic space which is separated and étale over a scheme is a scheme. Let $S$ be a locally noetherian scheme. I am looking for a reference giving an Artin's ...
0
votes
1answer
106 views

Picard group of a quotient of a group by its maximal parabolic subgroup

Let $G$ be a connected, linear, semi-simple algebraic group over an algebraically closed field of characteristic zero and $P$ be the maximal parabolic subgroup. We know that the quotient $Z=G/P$ is a ...
0
votes
0answers
63 views

Functor of order $n$ in Mumford's abelian variety

Let $T$ be a contravariant functor on the category of complete varieties into the Category $\underline{\mathrm{Ab}}$ of abelian groups. Let $X_0,\ldots,X_n$ be any system of complete varieties, ...
0
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0answers
115 views

Independent Generic Curves in the Projective Plane

I'm trying to read a paper by Masayoshi Nagata (available here) where he gives a counter-example to Hilbert's fourteenth prolem and I've run into some trouble understanding the terminology he's using. ...
0
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0answers
147 views

Different proof's of Marten's theorem

I am referring to Marten's theorem on the dimension of $W_d^r $ as in ACGH p. 192 . It seems to me that an even shorter proof can be given using Hopf's Theorem that if $\nu : A \otimes B \to C $ is ...
1
vote
0answers
170 views

What is deforming this non-complete intersection like?

Let $R = \mathbf{C}[x,y,u,v]$ be the coordinate ring of $\mathbf{C}^4$. Let $I$ be the ideal generated by $u$ and $v$, let $J$ be the ideal generated by $u$ and $y$. What are the flat deformations of ...
0
votes
0answers
93 views

Meromorphic functions on $U^2 = T^3 + 1$, genus [closed]

This question Asked in S.E but no, answer ,I would like to know how do i find a genus of $F$ . Let $k$ be a field of characteristic $\neq 2$, and consider the quadratic extension $F$ of $k(T)$ ...
2
votes
0answers
167 views

Can estimate upper bound of $|p_{i}|$ or $|q_{i}|?$

when I Find the diophantine-equation rational points $$2y^2=x^6-x^2+2$$ I using Faltings's theorem showed that there are only finitely many solutions,if we assmue that ...
-2
votes
0answers
45 views

Clockwise sorting of circle point [closed]

I have list of 3d points ( -2.03591339559,-0.560307972035,-0.474112849094), ( -2.05118196203,-0.55785528461,0.5743518821), ( -1.02999710644,1.16145402736,0.585203882893), ( ...
0
votes
0answers
73 views

A question about Segre class

Suppose $C$ is a cone over $X$.(i.e.$C=\operatorname{Spec}S$, where $S$ is a sheaf of $O_X$ algebras.) The Segre class $s(C)$of $C$ is the class in $A_*(X)$ defined by ...
0
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0answers
71 views

How can I keep the roots of f(x)^n+g(x)^m far away from the roots of f and g?

More specifically, suppose for example I have $h(x)=\sum_{i=1}^k (x-i)^{d_i}$. Can I get any handle on the roots of $h(x)$? Can I somehow guarantee that the roots of $h(x)$ are not arbitrarily close ...
6
votes
1answer
168 views

Decidability of an Algebraic System in Real Numbers

Is there an algorithm to decide whether an algebraic system \begin{gathered} {f_1}({x_1}, \ldots ,{x_n}) = 0 \hfill \\ \vdots \hfill \\ {f_m}({x_1}, \ldots ,{x_n}) = 0 \hfill \\ ...
0
votes
0answers
111 views

What is known about order of torsion of jacobian of hyperelliptic curve over finite field? [closed]

Suppose $J$ is jacobian of hyperelliptic curve $C$ over $F_p$ of genus $g$. Suppose $T$ is torsion of $J(F_p)$. What is known about order of $T$? Are there some bounds on order of $T$? Can one say ...
-1
votes
0answers
96 views

field of constants of a curve [closed]

I'm trying to gain some intuition about the field of constants of a curve. If $C$ is over a field $k$, then it is defined as the set of elements of $k(C)$ algebraic over $k$. If I understood ...
0
votes
0answers
125 views

Conditions for splitting of short exact sequence?

Are there conditions under which the short exact sequence $$0\rightarrow E (K)/mE (K)\rightarrow H^1_{Sel}(K,E_m)\rightarrow \Sha(E|K)_m\rightarrow 0$$ splits? I assume $K $ to be a number field and ...
14
votes
4answers
692 views

Number of $\mathbb F_p$ points constant mod $p$?

I have some affine varieties $X$ defined over $\mathbb Z$, and associated integers $c(X)$, with the property that $\# X_{\mathbb Z/p} \equiv c(X) \bmod p$ for all $p$. (In particular $c(X)$ is usually ...
1
vote
0answers
91 views

References for modular curves over finite fields [closed]

I'm looking for a detailed reference for modular curves over finite fields, such as $X(N)$, $X_1(N)$, and $X_0(N)$. There seems to be a lot of literature dealing with them over $\mathbb{C}$, but I'm ...
0
votes
1answer
102 views

rationality of residues of differentials

Let $C$ be a smooth curve over a field $k$, $\overline{C}$ the smooth compactification and $S=\overline{C} \setminus C$. We think of $S$ as a reduced divisor defined over $k$. Take the sheaf of ...
2
votes
2answers
214 views

Theta characteristics of genus$\geq3$ curve

Let $C$ be a smooth curve of genus$\geq3$ over $\mathbb{C}$, so there are $2^{g-1}(2^g-1)$ odd theta characteristics and $2^{g-1}(2^g+1)$ even theta characteristics. Do we know how many of them has ...
4
votes
2answers
306 views

Specialisation of rigid varieties

Recall that a variety $X$ over a field $k$ is called rigid if $H^1(X, T_X) = 0$. I am interested in understanding this property under specialisation. Let $R$ be a discrete valuation ring and let ...
7
votes
1answer
349 views
+200

Deformations of Ext rings

Let $k$ be a base ring and $k[x]$ the ring of polynomials in an indeterminate $x$ over $k$. Consider a (not necessarily commutative) algebra $A$ over $k[x]$ and two $A$-modules $M$ and $N$. Then for ...
1
vote
1answer
135 views

Morphisms contracting a family of curves

Let $f:X\rightarrow Y$ be a morphism of normal projective varieties. Let $S\subseteq X$ be a surface admitting a morphism $g:S\rightarrow C$ to a curve $C$ such that any fiber of $g$ is a curve. ...
1
vote
0answers
81 views

Looking for examples of holomorphic maps to $\mathbb{P}^1$ with certain property

I would like to know any example of nonconstant holomorphic map $f:X\to\mathbb{P}^1$ such that $K_X\cong f^*\mathcal{O}(2n)$ for some positive integer $n$, where $K_X$ is the canonical bundle of $X$. ...
3
votes
1answer
122 views

CM abelian varieties over the rationals

Let $K$ be a number field and let $A$ be an abelian variety of dimension $g$ over $K$. Let $L$ be a CM field and suppose that $[L:{\bf Q}]=2g$. Suppose that there exists an embedding ...
0
votes
0answers
34 views

Calculating the distinguished varieties of intersection product

In Fulton's Intersection theory Example 6.1.2,one considers two divisors on $\mathbf{P}^2$ given by $D_1=A+2B,D_2=2A+B$, where $A,B$ are lines meeting at a point. Let $X=D_1\times ...
0
votes
1answer
118 views

Sections of proper, flat morphism

Let $f:X \to Y$ be a proper, flat morphism of projective scheme and $Y$ is an irreducible, non-singular surface. Assume further that there exists a Zariski open subset $U$ of $Y$ whose complement is ...
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0answers
48 views

open set of hyperplanes not meeting a family of lines

Let \begin{align} \Omega=\begin{bmatrix} L_1 & \cdots & L_{n-1} \\ M_1 & \cdots & M_{n-1} \end{bmatrix}\end{align} be a matrix of linear forms on $\mathbb{P}^n$, i.e. homogeneous ...
3
votes
1answer
178 views

The stack of group algebraic spaces

The fibred category $\mathcal A$ of algebraic spaces over a scheme $S$ is a stack (over the category of affine schemes with the etale topology). This is proved in Laumon and Moret-Bailly's book (see ...
2
votes
0answers
66 views

Surjectivity locus of a morphism of families of sheaves

Let $X$ and $T$ be schemes and assume we have two coherent sheaves $\mathcal{F}$ and $\mathcal{G}$ on $X\times T$ which are flat over $T$, that is these are families of sheaves parametrized by $T$. ...
0
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0answers
74 views

a question about Nakayama functor [closed]

Assume $A$ a finite dimesional algebra over $k$, we assume $k$ algebraically closed, then can we computer $Hom_k(RHom_A(Hom_k(A,k),A),k)$ which is $N^2(A)$, here N is the derived Nakayama functor. For ...
0
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0answers
61 views

a closure argument regarding certain linear functionals on polynomials

Let $S_d$ be the vector space of homogeneous polynomials of degree $d$ in two variables $x,y$ over an algebraically closed field $k$. Let $\phi \in S_d^*$ be a linear functional on $S_d$, such that ...
3
votes
1answer
169 views

Stacks with a small coarse moduli space

Let $k$ be a field of characteristic zero. Let $X$ be a finite type algebraic stack over $k$ with a coarse (or good) moduli space $M$. Suppose that $M$ is isomorphic to a point, i.e., $M = Spec k$. ...
0
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1answer
138 views

Torsors and Central Extensions

In the setting of algebraic groups: I understand that a central extension of a group $G$ by an abelian group $A$ is a exact sequence of groups :$0\rightarrow A\rightarrow \tilde{G}\rightarrow ...
5
votes
1answer
237 views

Order of vanishing of an integer polynomial at a point

Let $f(x,y)$ be a polynomial with integer coefficients, and let $\alpha=(\alpha_1,\alpha_2)\in \mathbb{C}^2$ be a complex point. I want to show that $f$ cannot vanish at $\alpha$ to high order unless ...
2
votes
2answers
146 views

Pseudo-decision procedures for first order arithmetic

I was reading this paper http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.117.2911&rep=rep1&type=pdf in which the author describes an algorithm, based on Groebner basis, ...