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

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6
votes
1answer
194 views

Del Pezzo surfaces of degree $2$

I'm trying to understand the relationship between the different models of del Pezzo surfaces of degree $2$. Let $k$ be a field of characteristic not equal to $2$. Usually, del Pezzo surfaces of ...
4
votes
0answers
114 views

Relations between two definitions of non-archimedean analytic spaces

I begin to learn some non-archimedean geometry recently, and find that there are two different definitions of analytic spaces in the literature. Let us fix a non-archimedean complete valuation field $...
2
votes
0answers
72 views

GKZ decomposition for spherical varieties

If $X$ is a complete toric variety the GKZ decomposition of the effective cone $Eff(X)$ of $X$ corresponds to its Mori Chamber Decomposition, and therefore it encodes the birational geometry of $X$. ...
3
votes
1answer
271 views

Understanding an application of Riemann-Roch in an article

I saw the following in an article: Let $C$ be an irreducible smooth projective curve over an algebraically closed field $K$ and let $g$ be its genus. By Riemann-Roch, if N is large enough for every ...
3
votes
0answers
197 views

Equivariant sheaves over affine schemes

Let $k$ be a field, let $G$ be a linear algebraic group over $k$ and let $A$ be a commutative $k$-algebra which is acted on by $G$. We say that an $A$-module $M$ is a $(G,A)$-module if it satisfies ...
2
votes
1answer
175 views

Chow ring of an algebraic group for another equivalence relation than rational

For $G$ a split algebraic group of arbitrary Dynkin typ, the Chow ring with rational equivalence and $\mathbb{Z}/p\mathbb{Z}$, for $p$ some torsion prime of $G$, is well known and will be denoted as ...
3
votes
1answer
294 views

“Polygons and gravitons” and Kodaira's theorem

I'm trying to understand the paper by Hitchin called: ''Polygons and gravitons". I'm stuck at page 470. At this point, he does some computations and obtains the conformal structure of the real ...
6
votes
1answer
483 views

Is this a semi algebraic set?

Put $$A=\{(a_{0},a_{1},\ldots,a_{n}) \in \mathbb{C}^{n+1}\mid p(z)=a_{0}+a_{1}z+\ldots a_{n}z^{n} \;\;\text{is a one-to one function on the unit disc} \{z\in \mathbb{C} \mid |z|\leq 1\}$$ Is $\{(...
1
vote
1answer
172 views

Is the toric variety associated to this fan a weighted projective space?

Consider the complete fan $\Delta$ in $\mathbb R^2$ with edge vectors $v_1=e_1$ , $v_2=-a_1e_1+a_2e_2$ and $v_3=-b_1e_2-b_2e_2$ where $a_1,a_2$ and $b_1,b_2$ are respectively relatively prime positive ...
3
votes
2answers
423 views

These rings are isomorphic?

Consider the following rings: $A=\mathbb{C}\lbrace x,y,u \rbrace /(xy+x^3,y^2,xy^2+x^5) \ $ and $B=\mathbb{C}\lbrace x,y,u \rbrace /(xy+x^3,y^2+ux^4,xy^2+x^5)$ There is an isomorphism of $\mathbb{...
1
vote
1answer
147 views

simple normal crossing divisors on Fano manifold

Let $M$ be a Fano manifold. And $D=\mathop\sum\limits_{i=1}^r\tau_iD_i\in|-\lambda K_M|$ is a simple normal crossing $\mathbb{R}$-divisor where $\tau_i\in(0,1)$. Can we know that $D_i$s are ample (or $...
2
votes
2answers
153 views

Kähler forms arising as the curvature form of a singular metric on a line bundle

The Fubini-Study metric on complex projective space $\mathbb{P}^n$ is a smooth metric $h = e^{-\phi}$ on the line bundle $\mathcal{O}(1)$ and it is a standard calculation to check that its curvature ...
4
votes
0answers
130 views

Surjectivity of some evaluation map on global sections of a positive vector bundle

Let $X$ be a smooth complex projective manifold, let $E \rightarrow X$ be a Hermitian vector bundle and let $L \rightarrow X$ be a positive Hermitian line bundle. Let $H^0(X,E \otimes L^d)$ denote the ...
2
votes
0answers
60 views

Can we write an element in a super Grassmannian as a pair of matrices?

Super Grassmannians are introduced by Manin, see for example. Elements in a grassmannian can be written as matrices, see for example. Can we write an element in a super Grassmannian as a pair of ...
8
votes
1answer
254 views

Vanishing of Kahler differentials vs. surjective Frobenius?

Let $A$ be an $\mathbf{F}_p$-algebra such that $\Omega_{A/\mathbf{F}_p}=0$. Is the Frobenius map on $A$ surjective? Some context: i. The converse is clearly true. ii. The answer is yes if $A$ is a ...
13
votes
1answer
282 views

IC sheaf of certain explicit variety

Let $n,m$ be two positive integers. Let $Z$ denote the closed subvariety in $\mathbb A^n \times \mathbb A^m$ given by the equation $x_1...x_n=y_1...y_m$. QUESTION: What is the stalk (with the action ...
3
votes
1answer
205 views

Are rational surface singularities $\mathbb{Q}$-Gorenstein?

I know that, in general, rational singularities are not necessarily $\mathbb{Q}$-Gorenstein. So I ask: is there any positive result in this direction known for surfaces?
4
votes
1answer
1k views

Blow-up in family

Let $\pi \colon X \to T$ be a flat projective morphism, and let $Y$ be a closed sub-scheme of $X$ which is flat over $T$. We can assume that everything is defined over the complex numbers, and $T$ is ...
7
votes
1answer
335 views

Quasi-split tori and algebraic groups

Let $k$ be a perfect field. Recall that an algebraic torus $T$ over $k$ is called quasi-split if there exists some finite étale $k$-algebra $A$ such that $$T \cong \mathrm{R}_{A/k} \mathbb{G}_m.$$ A ...
3
votes
1answer
178 views

Tate modules of elliptic curves with complex multiplications

Let $E/K$ be an elliptic curve with complex multiplication over an imaginary quadratic field $K$. Then, I heard that it is well-known that the Tate module $V_{p}(E)$ over $\mathbb{Q}_{p}$ ...
4
votes
1answer
145 views

Computation on an Euler character

In the Bridgeland's paper "Flops and derived categories" (proof of (4.6), page 12), he computed an Euler character without much explanation. I thought this might not be difficult (and might not be ...
0
votes
1answer
114 views

Products of varieties of index 1

Let $k$ be a field of characteristic 0 and let $X$ and $Y$ be smooth, projective and geometrically integral $k$-schemes of finite type. Assume that both $X$ and $Y$ have 0-cycles of degree 1. Does $X\...
8
votes
1answer
371 views

Polynomial approximations of curves

This is the 3D version of this question. The responses to that question contained a lot of complaints about fuzzy definition of the problem, so I made this new question very narrow and explicit. For ...
5
votes
3answers
507 views

References - Voevodsky motives are the derived category of Nori motives?

First I would like to know if this has been worked out, and if the answer is affirmative I would like to know some references.
1
vote
0answers
89 views

Do we have super Plucker relations for a super Grassmannian?

Super Grassmannians are introduced by Manin, see for example. We have Plucker relation for Grassmannian. Are there some references about super Plucker relations for super Grassmannian? Thank you ...
2
votes
2answers
200 views

Is the zero locus of a global section flat?

Let $f:X \to Y$ be a surjective, smooth projective morphism of noetherian schemes. Let $\mathcal{L}$ be an inverible sheaf on $X$ satisfying $f_*\mathcal{L}$ is locally free and $s \in H^0(\mathcal{L})...
4
votes
2answers
251 views

Global section of universal bundle on Grassmanian

Let $G=G(k, V)$ be the Grassmanian of $k$-dimensional subspaces of the $n$th dimensional vector space $V$, regarded as a smooth algebraic variety over $\mathbb{C}$. Denote with $S$ the tautological (...
1
vote
0answers
74 views

functions coming from a perverse sheaf

Let's take a scheme $X$ over a finite field $k$ and $f:X(k)\rightarrow\mathbb{Q}_{\ell}$ What kind of condition do I need on $f$ if I want that it comes from an irreducible perverse sheaf on $X$?
2
votes
1answer
150 views

Universal family of grassmannian as projective bundle over $\mathbb P^n$

Let $p:X=\mathbb P(\mathcal E_2)\rightarrow Gr(2,n+1)$ be the universal family of the lines in $\mathbb P^n$. If we denote $e:X\rightarrow \mathbb P^n$ the natural projection, we have $\mathcal O_{\...
3
votes
1answer
178 views

Geometric contractibility of noetherian rings

Let $A$ be a noetherian ring. Let us define $A$ to be $n$-contractible if All locally free sheaves of rank $\le n$ over $\text{Spec} A$ is trivial. There exist a non-trivial locally free sheaf of ...
1
vote
1answer
142 views

Integral points - monotone symplectic toric manifolds

Suppose I am given a symplectic toric manifold $(M,\omega,\psi)$ which is also monotone, hence the symplectic form can be rescaled so that $c_1=[\omega]$. Then the moment map can be taken so that its ...
2
votes
1answer
98 views

Reference request for general Hurwitz spaces

Let $G$ be a fixed finite group. I'm interested in the structure of the set $\mathcal{H}_{r,g,h,G}$ of tuples $(C,f,\delta)$, where $C$ is a smooth projective genus $g\geq 2$ curve, $\delta:G\to\mbox{...
0
votes
0answers
28 views

Reducing certain monomials away from a set of polynomial equations

Suppose I have six polynomials $F_1, G_1, F_2, G_2, F_3, G_3 \in \mathbb{Q}[x_1, ..., x_n]$. The exact situation I have is the following: We have $\deg f_i = \deg G_i = d_i$ and $d_3 > d_2 > d_1$...
8
votes
0answers
299 views

How do I complete the sketch of proof in 'FGA explained', 5.5.8?

It seems to me that the one part that is difficult to transfer to general coherent sheaves is given only one sentence: "Because we have such a common $m$, we get as before an injective morphism from ...
5
votes
0answers
91 views

Is a Kummer surface over an finite field $\mathbb{F}_q$ supersingular iff $\mathbb{F}_q$-unirational?

Let $A$ be an abelian surface over an finite field $\mathbb{F}_q$. In particular, I am interested in the case when $A$ is a Jacobian variety. Is the Kummer surface $K_A/\mathbb{F}_q$ Shioda-...
2
votes
0answers
193 views

Hypersurfaces covered by high genus curves

Let $X$ be a smooth projective variety. The first question is the following: Is there a standard name or notation for the notion: the smallest integer (denoted by $g_{cov}(X)$ for now) such that ...
1
vote
0answers
113 views

Can an algebraic function be zero both at $z=0$ and at its leading singularity?

Apologies for asking possibly strange questions, but I am just a poor computer scientist trying to understand a mathematical paper on singularity analysis of algebraic functions that is apparently not ...
3
votes
0answers
42 views

Connectedness of semi algebraic set by c.a.d

I do not know whether there is a standard or some traditional ways to decide whether a semi algebraic set is connected or not. One way I know is c.a.d algorithm. I have read some papers of c.a.d ...
0
votes
0answers
128 views

Blowing up a subvariety.. what happens to the pullback of a divisor

Probably my question will sound a bit trivial.. but I know very well what happens in general by blowing up points, but here my question concerns with blowing up subvarieties in general. Suppose we ...
0
votes
0answers
104 views

Normalizer of non-split tori

Let $\mathbb{G}$ be a connected reductive group over $\mathbb{C}$. Let $G:=\mathbb{G}(\mathbb{C}(\!(t)\!))$. Let $T$ be a maximal torus in $G$. Question: What do we know about the normalizer $N_G(T)$...
3
votes
0answers
100 views

Non-multiplicative Euler-Poincaré Characteristics

Are there known examples of a non-multiplicative Euler-Poincaré characteristic on varieties? Let $\mathbf{Var}/k$ be the category of varieties over a filed $k$, i.e. the category of reduced separated ...
2
votes
0answers
63 views

Characters on lattices and isogenies of Abelian varieties

Let $V:=\mathbb{C}^g$ and $\Lambda \subset V$ be a lattice, i.e. a discrete subgroup of rank $2g$. Then $A:=V/ \Lambda$ is a complex torus of dimension $g$. We moreover assume that $A$ is algebraic, ...
10
votes
1answer
257 views

Is there a divisor in $\mathbb P^2$ such that all analytic maps into its complement algebraize?

Is there a closed subscheme $D$ in $\mathbb P^2_{\mathbb C}$ pure of codimension one such that, for all algebraic varieties $X$ over $\mathbb C$, any analytic map $$ \phi: X(\mathbb C) \to \mathbb P^...
3
votes
1answer
226 views

Finite generation of global sections of an invertible sheaf on a quasi-projective scheme

Let $X$ be a projective scheme over a noetherian ring, $\mathcal F$ an invertible sheaf on $X$, and $U$ an arbitrary open subset of $X$. Is $\Gamma(U,\mathcal F)$ a $\Gamma(U,\mathcal O_X)$-module of ...
0
votes
1answer
150 views

Normal bundle of a fiber of the family of curves

If we have the family of complex curves $f:X\rightarrow Y$, over a complex smooth curve $Y$ , we consider a fiber $C=f^{-1}(y)$ and its tangent bundle $T_{C}$. We know that $df: f^{*}{T_{Y}}_{|C}\...
1
vote
0answers
82 views

representability of a certain extension of group algebraic spaces

Let S be a scheme. Suppose we have sheaves in abelian groups $A,B,C$ over the big étale site of $S$. Suppose that $A$ and $C$ are representable by algebraic spaces in groups locally of finite type ...
8
votes
0answers
258 views

What is an example of a non-mixed $\ell$-adic sheaf?

$\def\FF{\mathbb{F}}\def\cG{\mathcal{G}}\def\QQ{\mathbb{Q}}\def\CC{\mathbb{C}}$I've been attending a reading seminar at Michigan on Kiehl and Weissauer's book Weil conjectures, perverse sheaves and l’...
2
votes
1answer
130 views

Transitivity of “being the zero locus of a section of a vector bundle”

Let $Z\subset Y\subset X$ be smooth projective $k$-varieties. Suppose $Z$ (resp. $Y$) is defined in $X$ by the vanishing of a section of a vector bundle $E$ (resp. $F$). Is it true that $Z\subset Y$ ...
4
votes
0answers
84 views

Can I combine the category of Drinfeld modules and the category of the base O_S

I am learning about Drinfeld modules,T-modules,...They are said to be analogues of elliptic curves, abelian varieties,... Let K be a finite extension of k = Frac(A), and $O_K$ the integral closure of ...
4
votes
0answers
230 views

étale cohomology of rings of integers of number fields and Shafarevich-Tate groups

Let $K$ be a number field, $A$ an abelian variety over $K$. Let $\mathcal{O}$ be the ring of integers of $K$, $\mathcal{A}$ the Néron model abelian scheme of $A$ over $\text{Spec}(\mathcal{O})$. For ...