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

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4
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239 views

questions about the “relative fundamental group” of SGA 1 Expose XIII

$\newcommand{\LL}{\mathbb{L}}$ I'm reading SGA 1, obtained from http://arxiv.org/abs/math/0206203 My questions regard 4.5 and 4.5.1 (page 309) of Expose XIII. Following "Exemples 4.4" in Expose ...
6
votes
1answer
331 views

Understanding of Tamagawa numbers of hyperelliptic curve

One's can find following definition of tamagawa numbers in Dino Lorenzini paper "Torsion and Tamagawa numbers": Let $K$ be any discrete valuation field with ring of integers $O_K$ , uniformizer ...
6
votes
0answers
106 views

Mumford curves for non-Schottky groups

I am not an algebraic geometer and I know next to nothing about moduli spaces, so the following question might be trivial, or well covered in the literature, only I didn't recognize it. So I am ...
3
votes
1answer
234 views

Reference for comparison of heart cohomology with standard cohomology

I'm looking for a reference for the following fact (which I believe to be true and should be easy for people who understand how spectral sequences arise from filtrations). Let A,B be two hearts of ...
0
votes
1answer
105 views

Big divisors and small transformations

Let $X$ be a smooth projective variety such that $-K_X$ is ample. Let $f:X\dashrightarrow Y$ be a small $\mathbb{Q}$-factorial transformation. I would like to know if is true or not that: $-K_Y$ is ...
15
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0answers
283 views

$\zeta(n)$ as a mixed Tate motive

I am trying to understand why there exists, for each $n \geq 2$, a mixed Tate motive $M$ over $\mathbb{Q}$ such that $M \in Ext^1_{MT(\mathbb{Q})}(\mathbb{Q}(0), \mathbb{Q}(n))$ and $\zeta(n)$, ...
4
votes
1answer
190 views

why Borel's computation implies Beilinson-Soulé?

Let $k$ be a field of characteristic zero and $DM(k)_{\mathrm{Q}}$ Voevodsky's category of motives over $k$ with rational coefficients. The Beilinson-Soulé conjecture says $$ ...
11
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2answers
616 views

Are there only finitely many associative algebras of fixed dimension?

Given an algebraically closed field $F$, for any positive integer $n$, are there always only finitely many non-isomorphic (noncommutative) associative algebras (possibly without identity) with ...
0
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0answers
94 views

How to prove a Lefschetzesque principle for relative homology and absolute homology

Here is my conjecture: If H is some homology theory from some abelian category A to some abelian category B, E is the class of all epimorphism in A, E' is some closed class of epimorphisms in A and ...
6
votes
1answer
91 views

Number of Richardson orbits in simple Lie algebras of types $E_n$?

This is a follow-up to my question about nilpotent orbits here asked in connection with an earlier discussion of symplectic resolutions. Leaving aside the connections with algebraic geometry and ...
7
votes
1answer
214 views

Extending holomorphic functions

Suppose $K\subset \mathbb{C}^n$ is a compact subset and $f:\mathbb{C}^n\setminus K\to \mathbb{C}$ is a holomorphic function. Then, provided $n>1$, $f$ extends to a holomorphic function defined on ...
1
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1answer
182 views

in which sense is a mixed Hodge structure an extension of pure ones?

I have heard several times that mixed Hodge structures are iterated extensions of pure ones. What does it mean? Here is what I figured out. A mixed Hodge structure $H$ comes with an increasing ...
4
votes
1answer
229 views

What goes wrong to use “bend-and-break” trick for singular varieties?

When $X$ is a smooth projective variety, one can use Mori's bend-and-break trick to establish the cone theorem. However, when $X$ has singularity (say klt. singularity), the cone theorem is obtained ...
6
votes
1answer
339 views

Elliptic curve and Galois representation

For an elliptic curve $E$ over ${\Bbb{Q}}$, let us consider Serre's mod $l$ representation by $\rho_{E,l} \colon {\mathrm{Gal}}({\overline{\Bbb{Q}}}/{\Bbb{Q}}) \to {\mathrm{Aut}}(\phantom{}_lE) = ...
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0answers
86 views

morphism of dual abelian variety

Let $A$ be an abelian variety over an algebraically closed field. My question is Dose there exist an ample line bundle $L$ on $A$, such that the morphism $f_{L}: A \longrightarrow A'$ defined by ...
2
votes
0answers
188 views

vanishing of étale cohomology of affine surface

Let $U$ be an affine smooth surface over an algebraic closure of a finite field. Let $\mathscr{A}/U$ be an Abelian scheme and $\ell \neq \mathrm{char}(k)$ be prime. Are there vanishing results for ...
23
votes
9answers
5k views

Why are polynomials so useful in mathematics?

This is perhaps unanswerable, or perhaps I am too algebraically ignorant to phrase it cogently, but: Is there some identifiable reason that polynomials over $\mathbb{C}$, $\mathbb{R}$, ...
6
votes
2answers
275 views

Calculate reduction of Jacobian of hyperelliptic curve

Suppose I have a hyperelliptic curve of genus $2$ over $\mathbb Q$. I want to get information about its Jacobian reduction at prime $p$ (especially, in case $p=2$). Also I'm interesting in the group ...
3
votes
1answer
187 views

Specialization Map of family of abelian varieties

In Lang's Survey on Diophantine Geometry, page 40, he said the following: Let $F=k(Y)$ be a function field of variety $Y$ over the constant field $k$ and $X_F$ a non-singular projective variety over ...
7
votes
1answer
272 views

Examples of Richardson orbit closures not having a symplectic resolution?

This is a follow-up to a recent question asked by Peter Crooks here. The answer by Ben Webster includes a helpful link to the corrected arXiv version of Baohua Fu's 2003 Invent. Math. paper ...
3
votes
2answers
221 views

Representing projective morphisms by homogeneous polynomials

Let $X$ be a projective variety over a field $k$ of characteristic $0$ and let $f: X \to \mathbb{P}^d$ be a $k$-morphism. Can we always find an embedding $i: X \hookrightarrow \mathbb{P}^N$ such that ...
0
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0answers
81 views

a reference for the fact that RQ/I is Gorenstein

I am looking for a reference to the fact that if R is Gorenstein, Q finite and acyclic quiver and I is admissible ideal, then RQ/I is also Gorenstein
3
votes
1answer
179 views

Unravelling some hypotheses on a variety

In Le group de Brauer II, Grothendieck states Proposition 1.4.- Soit $X$ a préschéma noetherien. Supposon que les anneaux hensélisés stricts des anneaux locaux de $X$ soient factoriels, [...] ...
6
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1answer
316 views

Motivation behind the proof that $X^4+Y^4+Z^4+W^4$ is unirational

I'm trying to understand the proof that $V(x^4+y^4+w^4+z^4)$ in $\mathbb P^3_k$ is unirational for $k=\overline{\mathbb F_3}$. The complete details are in the link so I just write a fast summary, ...
4
votes
1answer
164 views

Stronger versions of Schwartz-Zippel for random linear subspaces

This is a (self-contained) followup question to http://math.stackexchange.com/questions/380672/analogue-of-the-schwartz-zippel-lemma-for-subspaces. Let $f : \mathbb{R}^n \to \mathbb{R}$ be a nonzero ...
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0answers
135 views

Derived category of product of complex manifolds

Let $X$ and $Y$ be a compact complex manifold. Is it possible to describe the derived category $D^b(X\times Y)$ of coherent sheaves in terms of $D^b(X)$ and $D^b(Y)$? I am particularly interested in ...
0
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0answers
78 views

Regarding a paper relating surfaces and integrable mappings

This question has been posted on stackexchange, without answers. I'm reading the paper "A classification of two-dimensional integrable mappings and rational elliptic surfaces". I have two questions: ...
-2
votes
1answer
188 views

Schemes over $K_s$ and over $\bar{K}$

Let $K$ be a field. Let $X$ be a scheme over $K$. We denote by $K_s$ and by $\bar{K}$ the separable closure and the algebraic closure of $K$ respectively. By base change we have the schemes $X_{K_s}$ ...
1
vote
2answers
100 views

Degree of the Segre product of lines

Does anybody know whether there exists a proof by induction (or at least a proof that does not use Hilbert polynomials) that the degree of the Segre variety product of $n$ lines is $n!$ ?
2
votes
1answer
106 views

Smoothness of fix point components of finite group action on smooth variety

Let $X$ be a smooth complex algebraic variety, and $\varphi: \Gamma\curvearrowright X$ an action (by automorphisms) of a finite group $\Gamma$ on $X$. Can we say that each irreducible component of ...
3
votes
2answers
229 views

Algebraic Groups, Modules, and Comodules

Background: Let $H$ be a finitely generated commutative Hopf $k$-algebra, where $k$ is a field of non-zero characteristic. For $$ \widehat{H} := \text{Alg}_k\{H; k\}, $$ we recall (see Abe Chapter 4 ...
3
votes
2answers
300 views

Euler characteristics with and without compact support of algebraic varieties

Let $X$ be a complex algebraic variety, possibly singular and/or non-compact. It is well known that if $X$ is smooth then its Euler characteristic is equal to its Euler characteristic with compact ...
1
vote
2answers
108 views

kernel of isogeny becomes constant after base change

Let $S = Spec(O_K)$ be the spectrum of the rings of integers of a number field $K$. Let $A/S \setminus T$ be an Abelian scheme over an open subscheme $S \setminus T \subseteq S$. Does the kernel of ...
4
votes
1answer
113 views

Castelnuovo's rationality criterion on singular surfaces?

Let $S$ be a projective surface over an algebraically closed field. Suppose that $q(S)=h^1(\mathcal O_S)=0$ and $P_2(S)=h^0(\mathcal O_S(2K_S))=0$. If $S$ is smooth, Castelnuovo's rationality ...
1
vote
1answer
137 views

codimension one more than the dimension of a variety

If I have a variety $V$ with dimension $k$ and degree $d$, then a subspace of codimension $k$ in general position w.t.r. to $V$ intersects the variety in at most $d$ points. What can one say about a ...
3
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0answers
106 views

Non-vanishing of $H^0$ on rational surface

Let $X$ be a smooth rational surface that admits a proper birational morphism to $\mathbb{P}^2$ and $D$ a simple normal crossing divisor on $X$ such that $K_X+D$ is big, is it true that ...
3
votes
1answer
197 views

Why do we use analytic coordinates to characterize singularity?

I read about Du Val singularities on surface are classified by equations of ADE type. For example, $x^2+y^2+z^{n+1}=0$ for A type. As not every surface can have a neighbourhood embedded in ...
2
votes
2answers
290 views

Finite Quotients and Resolutions of Singularities

So, I feel like I'm missing something obvious, but I have the following situation: Let $X\to Y$ be a finite group quotient of schemes (in fact, varieties) by the finite group $G$. Let $\tilde{Y}\to ...
4
votes
0answers
158 views

Formal vs analytic trivializations of line bundles

Let $X$ be a smooth complex projective variety. Let $Y$ be a smooth divisor on $X$, and let $\mathfrak X$ be the formal completion of $X$ along $Y$. Question. If $\mathcal L$ is a line bundle on ...
5
votes
1answer
199 views

Which nilpotent orbit closures admit Springer resolutions?

Let $G$ be a connected, simply-connected complex semisimple group. We have the famous Springer resolution $$T^*(G/B)\rightarrow\mathcal{N}$$ of the closure of the regular nilpotent orbit. My ...
3
votes
3answers
533 views

Algebraic K-theory can be seen as a generalization of Linear algebra? [closed]

Algebraic K-theory can be seen as a generalization of Linear algebra? If yes, how so?
4
votes
1answer
429 views

A weak version of Bass' conjecture

Let $A$ be a finitely generated $\mathbb{Z}$-algebra which is a UFD. Then (a special case of) the Bass conjecture states that $K_0(A)$ is a finitely generated abelian group. As far as I am aware, this ...
2
votes
0answers
119 views

A generalization of the Weil reciprocity law in a case of any two sections of line bundles on a curve

It seems to me that there should exist a generalization of the Weil reciprocity law on curves, where instead of functions one takes arbitrary sections of two line bundles. More precisely, it might ...
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1answer
154 views

Some questions related to Iwasawa invariants of elliptic curves

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at an odd prime $p$. Let $\mathbb{Z}_{p}$ denote the ring of $p$-adic integers, and $\mathbb{Q}^{cyc}$ be the ...
3
votes
1answer
212 views

Conditions for a parametric curve to avoid self-intersection?

Suppose a planar curve $C$ is defined by parametric equations in $t$: $x(t)$ and $y(t)$. Are there conditions on these two functions that guarantee that $C$ does not self-intersect? For example, the ...
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vote
0answers
146 views

Coarse moduli spaces and rational points [closed]

Let $K$ be a field (not necessarily algraically closed). Let $\mathcal{F}$ be a contravariant functor from the category of schemes over $K$ to sets and $M$ be a coase moduli space for the functor. So, ...
0
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0answers
87 views

torsors on quasi-split groups

Let $\mathbf{G}$ be a split connected reductive group scheme over a scheme $X$. Let $X'\rightarrow X$ an étale Galois cover of group $\Gamma$. We consider $G$ a quasi-split group scheme over $X$ ...
2
votes
2answers
268 views

Local factors of Hasse-Weil zeta function - what do they have in common?

Let $X$ be a regular scheme, flat and of finite type over $Spec(\mathbb{Z})$ (add "projective" if you want). Then the Hasse-Weil zeta function of $X$ is defined as a product over all prime numbers of ...
2
votes
1answer
224 views

Are Abelian varieties (sometimes) globally $F$-split?

As defined by Karen Smith here, beginning of section 3? If $E$ is an elliptic curve, then it is when $E$ is ordinary. I wonder about higher dimension cases. Any references would be greatly ...
12
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1answer
487 views

Curves on K3 and modular forms

The paper of Bryan and Leung "The enumerative geometry of $K3$ surfaces and modular forms" provides the following formula. Let $S$ be a $K3$ surface and $C$ be a holomorphic curve in $S$ representing ...