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

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10
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0answers
269 views

Singular curve on an abelian surface

Let $C_2$ be a smooth genus $2$ curve and $J(C_2)$ its Jacobian. It is well known that the blow-up of $J(C_2)$ at the origin $o$ is isomorphic to the second symmetric product $\textrm{Sym}^2(C_2)$, ...
5
votes
1answer
185 views

Question on period map, Gauss-Manin connection and complex coordinates of $\mathcal{H}^1(k)$

Let $\mathcal{L}_g$ be the space of abelian differentials on Riemann surfaces of genus $g\ge 2$ and $\mathcal{TH}_g:=\mathcal{L}_g/Diff_0^+(S_g)$ be the Teichmuller space of abelian differentials on ...
2
votes
0answers
86 views

Behaviour of an étale morphism under Galois action on points

Consider the following situation. Let $k$ be a characteristic $0$ field, and consider an étale morphism of $k$ schemes $f:X\rightarrow Y$. Moreover, let $K$ and $L$ be two extension fields of $k$ such ...
-1
votes
0answers
121 views

Homotopy equivalence and sheaf cohomology

I have an inclusion $Y \hookrightarrow X$ of varieties that is a homotopy equivalence. ($X$ is a toric variety, $Y$ is a hypertoric variety, in case that is important) I know $H^0(\mathscr{O}_X)$, ...
1
vote
0answers
71 views

derived invariants, perversity and modular coefficients

Let $\pi:X\rightarrow Y$ a Galois cover of finite type schemes over $\mathbb{C}$ of group $\Gamma$. Let $n$ an integer such that it is not prime with the order of $\Gamma$. Then $\pi_{*}\mathbb{Z}/n\...
2
votes
0answers
189 views

Characterizing flatness in terms of pulling back relative schemes

All schemes are assumed to be noetherian (and maybe separated just in case) Here is a (possible) definition of flatness: Definition 1 (flatness): A morphism of schemes $f: X \to Y$ is flat iff ...
0
votes
1answer
59 views

representing base changes of the unit section

Let $S$ be a scheme and $G$ be a sheaf in groups on the big étale site over $S$. Let $e:S\rightarrow G$ be the unit section. Is it true that given an algebraic space in groups $H$, étale over $S$, and ...
1
vote
0answers
98 views

Isomorphism passing to the derived category

Suppose to have an additive right exact functor $F: \mathcal A \rightarrow \mathcal B$ between Abelian categories and suppose that $F(A)=B$ for an object $A$ in $\mathcal A$. Denote with $D(\mathcal A)...
1
vote
1answer
123 views

Possible `surgery' via formal neighborhoods

Let $ X\rightarrow Spec(\mathbb{C}[t])=\mathbb{C}$ be a projective variety over $\mathbb{C}[t]$ (a flat family of projective varieties $X_{t}$, $t\in \mathbb{C}$), and $X_{\eta}$ be the base change ...
4
votes
1answer
107 views

Cohen-Macaulay non-normal toric variety

Given a quasi-smooth toric variety $X$ in the sense of Gelfand-Kapranov-Zelevinsky, i.e. a (not necessarily normal) toric variety $X$ whose normalization is a $\mathbb{Q}$-factorial toric variety and ...
2
votes
0answers
86 views

Riemann-Roch Space for quotient curve

Let $C$ be a curve defined over a finite field $\mathbb{F}_q$. Let $\{f_1,..f_m\}$ be a basis for the Riemann-Roch space of functions, $\mathcal{L}(D)$, for the divisor $D= t\infty$. Suppose we have a ...
3
votes
1answer
109 views

ample subsheaf contained in the tangent bundle of projective space

Let $\mathcal F$ be an ample subsheaf of $T_{\mathbb P^n}$. Is it actually locally free? If not, is there a counterexample?
4
votes
0answers
73 views

Finding two hypersurfaces of the same degree that intersect $X/\mathbb{F}_q$ smoothly

Let $X$ be a smooth projective variety over a finite field. In [Poonen - Bertini theorems over finite fields] it is shown that one can find a smooth geometrically integral hypersurface $S$ of degree $...
1
vote
0answers
106 views

Gravitational instantons metric (change variables)

I'm trying to understand the paper by Hitchin called: ''Polygons and gravitons". I'm stuck at page 471. At this point, he does some computations and obtains a metric: $$ \gamma dz d\bar{z}+\gamma^{...
5
votes
1answer
608 views

Ordinary primes vs supersingular primes

Let $E$ be an elliptic curve over $\mathbb{Q}$ without CM. As shown by Serre, the set of supersingular primes for $E$ has density zero. Is the analytic rank of $L(E,1)$ determined only by the ...
0
votes
1answer
176 views

Proj of some graded algebra

I was computing some GIT quotients and came up with the following question: to compute $\mathrm{Proj}(\mathbb C [f_1,f_2,f_3,f_4,f_5,f_6]/I)$ where $f_i$'s are homogeneous polynomials of same degree ...
3
votes
0answers
114 views

Self-intersection of sum of Eff cone generators on Picard rank 2 surfaces

Let $S$ be a smooth, projective, complex surface with Picard rank 2, whose effective cone is generated by two curves of negative self-intersection, $C_1$ and $C_2$ (i.e. $C_1^2<0$ and $C_2^2<0$)....
0
votes
2answers
315 views

The non-existence of the fine moduli scheme of vector bundles. Why?

The reference I am using is this one. The question is about the moduli space of vector bundles. I am trying to understand why the fine moduli scheme does not exist. Let $C$ a projective curve. Let $S$ ...
0
votes
1answer
103 views

Sheaf of relative differentials of double cover

Let $Y$ be a smooth projective $k$-variety, $D\subset Y$ a smooth (irreducible) divisor and a line bundle such that $L^2=\mathcal O_Y(D)$. Let us call $f:X\rightarrow Y$ the double cover defined by ...
2
votes
1answer
147 views

General classification of degree $d$ homogeneous polynomials in $\mathbb{P}^2(\text{finite field})$?

As the question title suggests, what is the general classification of degree $d$ homogeneous polynomials in $\mathbb{P}^2(\text{finite field})$, up to projective equivalence?
1
vote
1answer
185 views

Direct image of a vector bundle under birational morphism

Let $f:X\rightarrow Y$ be a birational map of smooth projective varieties over complex numbers. Let $E$ be a vector bundle on $X$. Will $f_*E$ be a reflexive sheaf. Is it possible to impose some ...
3
votes
0answers
162 views

Why define curves over perfect fields?

One may define a curve (e.g. separated scheme of finite type of dim. 1) over an algebraically closed field, as done in Hartshorne's book. A weaker assumption, which is used commonly, is to define a ...
1
vote
0answers
42 views

Sieving for points not on a conic and quadratic residues

Let $$\displaystyle F(x,y,z) = \sum_{\substack{0 \leq i_1, i_2, i_3 \leq 2 \\ i_1 + i_2 + i_3 = 2}} a_{i_1, i_2, i_3} x_1^{i_1} x_2^{i_2} x_3^{i_3}$$ be a ternary quadratic form with integer ...
4
votes
0answers
122 views

Centralizer action on components of Springer fibers

Let $G$ be a complex adjoint group. Let $u\in G$ be unipotent. The group $A(u):=\pi_0(Z_G(u))$ acts on the set of components of the Springer fiber $\mathcal{B}_u$, the variety of Borel subgroups that ...
-1
votes
0answers
61 views

intersection between conic and cubic [migrated]

Let $ A=V(\alpha)$ a non-singular conic in $\mathbb{P}^2(\mathbb{C})$, $V(\delta_1)$ and $V(\delta_2)$ two cubic such that : $D_1$ and $D_2$ meet $A$ tangentially at six distincts points $P_1,...,P_6$ ...
11
votes
2answers
370 views

Affine GIT is an open map?

Let $k$ be a field, $X= \text{Spec}\,A$ be an affine scheme, with $A$ a finitely generated $k$-algebra. $G=\text{Spec}\,R$ is a linearly reductive group acting rationally on A, i.e. every element of $...
1
vote
0answers
107 views

Blow up and strict transform

Let $X$ be a smooth projective variety over $\mathbb{C}$. Consider the blow up of $X$ about a closed subvariety $Z$. Let $X'=Bl_Z(X)$. Let $Y$ be a smooth irreducible divisor of $X$ properly ...
1
vote
0answers
93 views

On dimension of the moduli space of abelian differentials on Riemann surfaces

I fear I'm missing something important here, so forgive me if my question is stupid. Consider $\mathcal{M}_g$ the moduli space of Riemann surfaces of genus $g>2$ and $\mathcal{H}_g$ the moduli ...
0
votes
0answers
49 views

Toric divisor with respect to the face of polytope

As well-known in toric geometry, we can define a toric divisor $D_i$ with respect to every face $F_i$ of the polytope. And it is well know that $-K_M=\sum D_i$. My question is that if we denote $D_i$ ...
5
votes
2answers
233 views

When is $\mathbb C^d\setminus\mathcal Z$ simply connected?

Let $\Delta$ be a fan in the lattice $N\cong\mathbb Z^n$ with $d$ edges $\{\rho_1,\cdots,\rho_d\}$. Consider the co-ordinate ring $\mathbb C[x_1,\cdots,x_n]$. Let $\mathcal Z=\bigcup_C\mathcal V(x_i\ |...
4
votes
1answer
86 views

Is there a relation between the singularities and the divisor class group of a simplicial toric variety

Let $\Delta$ be a simplicial fan in the lattice $N\cong\mathbb Z^n$ with $d$ edges and $\{u_1,\cdots,u_d\}$ are the primitive vectors along the edges. Let $A$ be the divisor class group of the ...
4
votes
0answers
67 views

Morphisms of local nature for topologies that are the same

Let $S$ be a scheme. It is known that the smooth topology on $\textrm{Sch}_S$ is equivalent to the étale topology, basically because every smooth covering can be refined to an étale covering. ...
1
vote
1answer
213 views

General fiber of a rational map

Let $f:X\dashrightarrow Y$ be a rational map, where $X,Y$ are reduced and irreducible varieties over a field of characteristic zero. Is the general fiber of $f$ always reduced? Is this true if we ...
1
vote
2answers
159 views

How to prove that $A$ is supersingular iff the Picard number $\rho(A)$ is equal to the second $l$-adic Betti number $b_2(A) = 6$?

Let $A$ be an abelian surface over algebraically closed field $k$ of characteristic $p > 2$. How to prove that $A$ is supersingular (in other words, there is an isogeny between $A$ and $E^2$, where ...
1
vote
0answers
105 views

Is there a reference for boundedness of smooth canonically polarized varieties over Z (No…)

In Kollár's paper Quotient spaces modulo algebraic groups, Kollár mentions right above Theorem 1.8 that the stack $\mathcal M_P$ of smooth canonically polarized varieties over Spec $\mathbb Z$ with ...
4
votes
2answers
216 views

Reference request on birational invariance of Chow group of zero cycles of degree zero

Let $CH_0(X)^0$ denote the group of zero cycles of degree zero modulo rational equivalence. I am looking for a reference for the following fact: If $X$ and $Y$ are smooth and projective varieties ...
2
votes
1answer
281 views

Are there analogies between $\Bbb F_q[x_1,x_2]$ and a suitable object related to $\Bbb Z$?

Much progress in understanding $\Bbb Z$ is made from analogies between $\Bbb F_q[x]$ and $\Bbb Z$. Can there be analogies between arithmetic in $\Bbb F_q[x_1,x_2]$ and a suitable object related to $\...
0
votes
1answer
115 views

Clarification on the definition of a quotient singularity

I am working on the quotient construction of a simplicial toric variety as described in chapter 5 of this book. I have tried the following two examples - The fan $\Delta$ in $\mathbb R^2$ consists ...
0
votes
0answers
8 views

Why no prime number could appear as the length of a hypotenuse in more than one Pythagorean triangle? [migrated]

Why no prime number could appear as the length of a hypotenuse in more than one Pythagorean triangle? In other words, could any of you give me a algebraic proof for the following? given prime ...
6
votes
1answer
191 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
269 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
196 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
168 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
171 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 ...