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

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2
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0answers
57 views

Is it possible to assume that an ´etale neighborhood is connected?

I am new to ´etale topology (though I've seen Grothendieck's sites before). Let $S:=\mathcal{O}^\textrm{sh}_{X,x}$ be the strict local ring of a point $x$ of a scheme $X=\textrm{Spec}R$ (over a ...
2
votes
1answer
141 views

Automorphisms of ideals of $\mathbb{C}[t]$

Let $f(t)\in\mathbb{C}[t]$, and let $I_f$ be the ideal in $\mathbb{C}[t]$ generated by $f(t)$. The ideal $I_f$ has a natural $\mathbb{C}$-algebra structure. My question is the following: For ...
0
votes
0answers
89 views

Linearization of line bundle [on hold]

In the definition of the linearization of a line bundle in Dolgashev book [Lectures on Invariant Theory, page 104], there are two conditions : 1) the action of the group $G$ on the line bundle $L$ ...
5
votes
2answers
212 views

de Rham cohomology of smooth affine varieties

Let $U$ be a smooth variety over $\mathbb{C}$. We know that there exists a smooth compactification $X$ such that $X-U$ is a normal crossings divisor $D$ and that the de Rham cohomology of $U$ can be ...
2
votes
2answers
168 views

Fano manifold admit an smooth anti-canonical divisor?

Let $M$ ba a compact Kaehler Fano manifold. Under which conditions $M$ admit a smooth anti-canonical divisor $D$
18
votes
2answers
399 views

What is the purpose of the flat/fppf/fpqc topologies?

There have been other similar questions before (e.g. What is your picture of the flat topology?), but none of them seem to have been answered fully. As someone who originally started in ...
1
vote
0answers
92 views

Hodge structures generated by cohomology groups of varities with dimension less than $n$

Let $X$ be a smooth projective variety over $\mathbb{C}$ with dimension $n$. Is it true that for every $i<n$, the Hodge structure on $\mathrm{H}^i(X,\mathbb{Q})$ is generated by Hodge structures of ...
2
votes
2answers
215 views

cohomological obstructions and rational points

Let $X$ a (nice) scheme over $\mathbb{Q}$. Are there cohomolgical obstructions answering the following questions: 1) is $X(\mathbb{Q})$ an empty set ? 2) is $X(\mathbb{Q})$ a finite (non empty) set ...
0
votes
1answer
92 views

Alexeev's projective torus embeddings

I'm reading Alexeev's "Complete moduli in the presence of semiabelian group action" and I just started to study toric geometry. In chapter 2 in order to obtain an affine toric variety he takes ...
3
votes
1answer
133 views

Proof of a Proposition regarding the reduction of N-torsion groups on elliptic curves

In Diamond-Shurman A first course in Modular forms p.334 Prop. 8.4.4. It is stated, For E elliptic curve over $\bar{\mathbb{Q}}$ with good reduction at the prime ideal $\mathfrak{p}$ the reduction ...
3
votes
1answer
183 views

Is an anticanonical Weil divisor in $\mathbb{Q}$-Gorenstein variety Calabi-Yau?

Let $P$ be a normal, $\mathbb{Q}$-Gorestein variety with terminal singularities. Let $X \subseteq P$ be a normal, irreducible Weil divisor such that $X \sim_{\mathbb{Q}} - K_P$, that is ...
1
vote
0answers
51 views

pushforward of universal objects along canonical morphisms of stacks

The kind of question I'm interested in has the following flavor: having two moduli stacks with one being an enhanced (i.e more data) version of the other with the natural "forgetting" map between them ...
0
votes
1answer
103 views

Intersection Matrix of a resolution

Probably this is a very easy question. Let $f:X\rightarrow S$ be a resolution of a projective surface such that $$K_X = f^{*}K_S+\sum_ia_iE_i$$ with $a_i>0$. By Grauert-Mumford theorem the ...
0
votes
0answers
102 views

p-divisibility of the connected component of the Picard group [on hold]

Let $X$ to be a smooth projective variety over a field of positive characteristic $p>0$, then can one claim $Pic^0(X)$ is p-divisible.
2
votes
0answers
119 views

Hilbert vs Chow in nice cases

I'm trying to understand the relationship between the Hilbert schemes and Chow varieties in situations where everything is simple. Suppose that $X$ is a smooth projective variety over $\mathbb C$, ...
2
votes
0answers
85 views

Number of common solutions of polynomial system

Let $ \mathbb{F}_p$ be a finite field and $\{f_j\}_{j=1}^{j=r} \subseteq \mathbb{F}_p[X_1,...,X_n]$ be a set of polynomials. Let consider the system of equations: $f_j(x_1,...,x_n)=0$ for $j = ...
3
votes
1answer
168 views

line bundle descends?

Let the permutation group $S_4$ act on $\mathbb C^4$ by permuting the coordinates. Consider the categorical quotient $\mathbb P(\mathbb C^4)/S_4$. It is a projective variety by a theorem of Mumford. ...
1
vote
1answer
148 views

Explicit examples of Hasse--Weil zeta-function calculations for curves

The problem of calculating Hasse--Weil zeta-function for a given curve $C/\mathbb{F}_p$ over a finite field is far from being easy, especially for large genus (as discussed by Wouter Castryck at ...
2
votes
0answers
169 views
+50

Reference request for the relation of Ext groups and bar construction

I need a reference for the description of Ext groups in mixed categories (i.e. abelian categories with a weight filtration and semisimple graded quotients) by using the bar complex, as mentioned in ...
2
votes
0answers
132 views

A morphism of elliptic schemes that preserves the identity is a homomorphism

I am trying to understand the proof of the fact that any morphism $f \colon E_1 \rightarrow E_2$ of elliptic curves over an arbitrary base scheme $S$ satisfying $f(0) = 0$ must respect the group ...
4
votes
0answers
126 views

Inversion, Koszul duality, combinatorics and geometry

According to this MO answer Koszul duality is related to operations on generating series; 1) multiplicative inversion for quadratic algebras, 2) compositional inversion for quadratic operads, 3) ...
15
votes
2answers
344 views

Can all unit-distance graphs have their vertices at algebraic integers?

A graph $G$ is described as a unit-distance graph if there exists a function $f:G \rightarrow \mathbb{C}$ such that for every edge $(u,v) \in E(G)$, we have $|f(u) - f(v)| = 1$. Obviously, we can ...
8
votes
0answers
207 views
+500

Does a semistable curve descend to a regular base?

Let $f\colon X \rightarrow S$ be a semistable curve of genus $g \ge 0$. Being a semistable curve means that $f$ is a morphism of schemes such that $f$ is proper, flat, and of finite presentation; ...
3
votes
1answer
145 views

When does a hyperelliptic Riemann surface admit a map of degree 3

Let $X$ be a hyperelliptic curve of genus $g>1$. For which $g$ does $X$ admit a map $X\to \mathbb P^1$ of degree $3$? I think a genus two curve $X$ admits a map of degree $3$. Proof: Pick $P$ ...
9
votes
0answers
183 views

Do canonical stacks exist over Spec(Z)?

Suppose a scheme $X$ has tame quotient singularities. Does there exist a smooth DM stack $\mathcal X$ with coarse space $X$ so that the coarse space morphism $\mathcal X\to X$ is an isomorphism ...
2
votes
0answers
104 views

Invariant generalized sections of dual vector bundles

Assume X is a real smooth manifold with an action of the real Lie group G. Let E be a G-vector bundle over X. Consider the spaces of generalized sections over X of E, and of E^* (fiberwise dual). My ...
0
votes
0answers
87 views

Coefficients of Hilbert polynomials

Recall that we can define the Hilbert series of a graded commutative algebra $$\displaystyle S = \bigoplus_{n \geq 0} S_n$$ over a field $K$ by $$\displaystyle \mathcal{H}_S(t) = \sum_{n=0}^\infty ...
0
votes
0answers
48 views

Extending a model to a given compactification of its generic fiber

Let $R$ be a discrete valuation ring and $K$ its field of fraction. Let $X$ be a proper $K$-variety, $U$ a dense open and consider an $R$-model $\mathcal{U}$ of $U$. Can we embed $\mathcal{U}$ in a ...
3
votes
1answer
165 views

References for general Hasse-Weil zeta function

Most research on the Hasse-Weil zeta function focuses on some particular type of algebraic variety, and general surveys usually deal mostly with the better understood elliptic curve case. I am ...
0
votes
0answers
46 views

Complementary polynomials

Denote $S=\{0,1\}^n$. $\mathsf{MLP}_{d,n}=\{p\in\Bbb R[x_1,\dots,x_n]:p\mathsf{\mbox{ is mutilinear with total degree}}(p)=d\}$. Is there an $n\geq d^2+1$ such that there exists distinct polynomials ...
0
votes
1answer
94 views

Schematic image of a relative Cartier divisor of a fiberwise dense open

Let $S$ be a scheme and $A$ an abelian $S$-scheme, i.e., $A \rightarrow S$ is a proper smooth $S$-group scheme whose fibers are $g$-dimensional abelian varieties. Suppose that one has a fiberwise ...
5
votes
0answers
96 views

Groupoid cardinality of DM stack and point counting on coarse moduli spaces

Let $X$ be a finite type DM stack over a finite field $k$ with a coarse moduli space $X_c$. (We only assume $X_c$ is an algebraic space and $X$ might have infinite inertia stack.) Under which ...
0
votes
0answers
73 views

Moduli space of points - Gorenstein ideal

I've been working on algebraic covers, $\varphi\colon X\rightarrow Y$, ($\varphi_*\mathcal{O}_X$ is a locally free $\mathcal{O}_Y$-algebra of rank d). I'm more interested in the algebraic point of ...
2
votes
1answer
157 views

Difference between Frobenii on Tate modules of special and generic fibre

Let $E$ be elliptic curve over $\mathbb Q$ and $p$ a prime of good reduction for $E$. Fix $\ell \neq p$. If $E_p$ is ordinary then we have Frobenius $F_p$ on $E_p$. Assume $F_p$ lifts to ...
12
votes
1answer
508 views

Elliptic curves and connected components

Are there elliptic curves of positive rank with two real connected components in which all the rational points lie only on one component? Concrete examples are really appreciated.
0
votes
0answers
70 views

What does the equation $\tau \tau^* = \sigma^* \sigma$ represent in the ADHM construction of vector bundles?

I'm looking at the explicit construction of vector bundles with Anti-Self-Dual (ASD) connections on them via the ADHM construction of instantons. At the heart of this is the complex $$ V ...
-2
votes
0answers
88 views

Exercise from Geometry of algebraic curves by Arbarello, Cornalba, Griffiths, Harris [closed]

Let π:C′→C an unramified double cover of a complex Riemann surface C of genus g. With the symbol Nmπ we mean the norm application that takes a meromorphic function on C′ named h and produce a ...
-2
votes
0answers
68 views

involution of a riemann surface [closed]

I've have found in the book GEOMETRY OF ALGEBRAIC CURVES by Arbarello, Cornalba, Griffiths that if π:C′→C it's a double unramified cover of a complex riemann surface named C that we can define the ...
0
votes
0answers
58 views

Circle squarer and solution of polynomial equations [closed]

Given a circle squarer, i.e. something that gives a length of $\pi$ given an unit length and vice versa, and a straightedge and compass, is it possible to solve any polynomial equation with rational ...
4
votes
0answers
103 views

Global Affine Flag Variety and Affine Flag Variety

There is a construction of a global affine flag variety over $\mathbb{A}^1$ (or another curve) $Fl_{\mathbb{A}_1}$ such that each fiber above $\epsilon \neq 0$ is isomorphic to a direct product of the ...
2
votes
0answers
54 views

Does the invariant from resolution of singularities provide a Whitney stratification?

The topic of Whitney stratifications came up in a lecture, and the general procedure in the examples was to decompose the singular locus of the variety into the strata starting with the "worst" ones. ...
0
votes
0answers
143 views

What is a Beilinson spectral sequence?

I'm writing to ask just a question. I would like to understand better what is the Beilinson's spectral sequence and how it can be used. Is there any useful and clear reference you advice to someone ...
1
vote
1answer
292 views

Isomorphism between a mapping class group and the fundamental group of a moduli space

For some fixed integer $d \geq 3$, let $M(0, d)$ be the mapping class group of self-homeomorphisms of the Riemann sphere which fix each of the $d$ points $0, 1, ... , d-2, \infty$. Let $X$ be the ...
4
votes
1answer
153 views

Verlinde Formula and Theta Function Identities

The paper Fusion rules and modular transformations in 2D conformal field theory by Erik Verlinde mentions a simple case of rational conformal field theory, where the fusion algebra is just ...
2
votes
1answer
149 views

rational point of a curve [closed]

Let $X$ be a smooth projective curve over $\mathbb{Q}$. I heard (if I did not misunderstood) that the geometry of the complex points $X(\mathbb{C})$ (flat, hyperbolic case) dicts the shape (group ...
2
votes
2answers
197 views

Rigid curves, and the “richness” of their homology class

Let $X$ be a complex smooth projective variety, and $C\subset X$ a smooth curve. Then $C$ defines a cycle $$\beta=[C]\in H_2(X,\mathbb Z).$$ I have a very vague question about this situation: Q. ...
3
votes
1answer
179 views

Abelian varieties as analytic manifolds

Assume we have an Abelian varieties over the p-adic numbers, namely $ k=\mathbb{Q}_p$. Then the question is whether $A(k)$, the rational points over $k$, will form a p-adic analytic manifold. I am ...
-2
votes
0answers
63 views

Existence of general element in module case [closed]

Let $(R,m)$ be a noetherian local ring. Let $a_1,a_2,a_3\in R\oplus R$ and $S=R[X_1,X_2,X_3].$ Then is it true that if $z=a_1X_1+a_2X_2+a_3X_3\in S\oplus S$ then $\frac{S}{zS}\cong S[Y].$
10
votes
1answer
194 views

Is there a solvable point on any variety over the field of complex rational functions?

Let $K = \mathbb{C}(T)$ be the field of complex rational functions in one variable, and let $V$ be a variety defined over $K$. Must $V$ have a solvable point? The variety $V$ is assumed ...
6
votes
1answer
299 views

Étale cohomology versus classical cohomology

Let $X$ be an algebraic variety over $\mathbb{C}$. If $X$ is smooth, the étale cohomology $H^p_{\textrm{ét}}(X,\mathbb{Z}/n)$ is isomorphic to the singular cohomology ...