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

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366 views

Can we construct cohomolgy theory on noetherian separated schemes without Axiom of Choice?

The usual cohomology theory on schemes uses injective or flasque resolutions of quasi-coherent sheaves. Hence it uses Axiom of Choice. However, if the base scheme is a noetherian separated scheme, the ...
7
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0answers
88 views

Lifting Abelian Varieties to p-adic fields

Assume I have an abelian variety $A$ over a finite field $k$ of characteristic $p$. Work of Norman and Oort (1980) says I can lift $A$ to an abelian variety $\mathscr{A}$ over some characteristic ...
7
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1answer
208 views

The moduli space of special Lagrangian submanifolds

Given a special Lagrangian fibration $f:M \rightarrow B$ of a Calabi-Yau manifold $M$, one can associate to it two affine structures (symplectic and complex) on the base space $B$. A theorem of ...
9
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4answers
240 views

Is any quadric birational to a product of Brauer-Severi varieties?

Let $k$ be a field with algebraic closure $\bar k$. Assume that $k$ is perfect and not of characteristic $2$ for simplicity. Let $$X: \quad Q(x)=0, \quad \subset \mathbb{P}^n_k,$$ be a non-singular ...
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1answer
92 views

Bott formula for projective bundles

For a projective space one has Bott formula to compute $h^q(ℙ^n,Ω^p(k))$, where $Ω^p(k)$ is the k-twisted sheaf of sections in the p-th power of the cotangent bundle of $ℙ^n$. I am wondering if there ...
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1answer
97 views

Intersection theory on M_{g,n}

Is there a paper\book that lists the top intersections of Hodge classes and tautological classes on $\overline{\mathcal{M}}_{g,n}$ for small $g$ and $k$, e.g. $g=2,3$ and $k=0,1,2$ ?
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70 views

Veronese surface [on hold]

I have a question(Hartshorne ,page 13,exercise 13): If Y be the image of the 2-uple embedding(described in exercise 12) of P^2 in P^5. and if Z(a subset of Y) is a closed curve(variety of dim 1) show ...
9
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2answers
292 views

What are the invariants of $U\otimes V\otimes W$ under action of $GL(U)\times GL(V) \times GL(W)$

The tensor product of some (finite dimensional real) vector spaces is acted on by the direct product of their general linear groups. I would like to know if there are explicit invariants in the case ...
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1answer
82 views

Kaehler form on weighted projective space

The Kaehler potential for the standard Fubini-Study Kaehler form in projective space $\mathbb{C} P^n$ is given by: $$\log(\sum_{i=0}^n |z_i|^2)).$$ What is the analogous formula for a Kaehler ...
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2answers
239 views

Lifting a birational map of $X/G$ to a birational map of $X$?

Let $X$ be a projective complex manifold and $G$ a finite group. Assume that $G$ acts on $X$ holomorphically and freely. Is it true that any birational map $\phi \in Bir(X/G)$ lifts to some birational ...
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2answers
233 views

Effective divisors on the blow-up of $\mathbb{P}^n$

Let $X^n_r$ the blow-up of $\mathbb{P}^n$ at $r$ points in very general position. (1) It is known in general the Effective Cone or the Numerical Effective Cone of those algebraic variety? Let $X$ ...
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83 views

Simple example of isolated critical point with non-semisimple monodromy

Consider a polynomial map $f :\mathbb{C}^{n+1} \rightarrow \mathbb{C}$ with $f(0)=0$ (no constant term) and with isolated critical point at $0 \in \mathbb{C}^{n+1}$. We can choose a disc $D$ of some ...
5
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2answers
232 views

Equivariant derived category and invariant divisor

I'm looking for a reference of the following (folklore?) result. Let $X$ be a smooth projective variety equipped with a $G=\mathbb{Z}/2\mathbb{Z}$ action (we consider the simplest case, everything ...
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148 views
+50

Pullback of a sheaf associated to a divisor

I am reading a paper Desingularisation des varietes de Schubert generalisees by Demazure. I am interested in Lemma 3 on page 58. In particular, I would like to know whether the lemma is true and how ...
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67 views

Comparison of K-groups of (affine) singular schemes with K'=G-groups

It is well known that Quillen K-theory coincides with $K'=G$-theory for regular schemes, and can be distinct from it for singular ones. Are there any methods for studying this distinctions? In ...
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2answers
940 views

What is a Gaussian measure?

Let $X$ be a topological affine space. A Gaussian measure on $X$ is characterized by the property that its finite-dimensional projections are multivariate Gaussian distributions. Is there a direct ...
4
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1answer
137 views

Blow-up of $\mathbb{P}^4$ along a quadric surface

Let $Q\subset\mathbb{P}^3\subset\mathbb{P}^4$ be a smooth quadric surface, and let $X = Bl_Q\mathbb{P}^4$ the blow-up of $\mathbb{P}^4$ along $Q$. Let $H$ be the pull-back of the hyperplane section of ...
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77 views

Degree and quasi projective family

Let $V$ be a quasi-projective variety in $\mathbb{P}^{n}\times\mathbb{P}^m$. If $p\in \mathbb{P}^m$, we define the degree of $V_p$ as the degree of its closure in $\mathbb{P}^n$. Question : $\exists ...
5
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3answers
835 views

Is the inertia stack of a Deligne-Mumford stack always finite?

Let X be a DM stack over a field k. We follow the definition in Laumon and Moret-Bailly's book, so that its diagonal is quasi-compact (and hence diagonal is of finite type). Then is the diagonal ...
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44 views

Quasi-finite morphisms of stacks

Let $f:X\to Y$ be a morphism of ``nice" stacks over $\mathbf C$ such that the induced morphism on coarse moduli spaces is quasi-finite. Is $f$ quasi-finite? By a "nice" stack I mean a smooth finite ...
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46 views

plotting parametrized algebraic curves near singularities

I have a parametrized algebraic curve: x(t)=A(t)/D(t); y(t)=B(t)/D(t); with A(t) and B(t) being polynomials in t. The curve is solution of a linear system in two unknowns x and y with Cramer's ...
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1answer
232 views

Top self-intersection of exceptional divisors

Let $Y\subset\mathbb{P}^n$ be a smooth variety of codimension two. Consider the blow-up $X = Bl_Y\mathbb{P}^n$ of $\mathbb{P}^n$ along $Y$, and let $E$ be the exceptional divisor over $Y$. Then $E$ ...
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62 views

Local system over $\mathcal A_{g,[n]}$ with unipotent monodromy

Let $\mathcal A_{g,[n]}$ denote the moduli space of principal polarized abelian varieties with level-[n] structure and $\bar {\mathcal A}_{g,[n]}\supset \mathcal A_{g,[n]}$ a smooth Toroida ...
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2answers
312 views

Holomorphic trivialization of $(x,y) \subset \mathbb{C}[x,y]/(y^2 - x^3 + x)$

This question is largely out of curiosity but also motivated by an attempt to understand vector bundles on elliptic curves better. I believe it is a theorem of Grauert that any holomorphic vector ...
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3answers
1k views

Algebra and Cancer Research

Let me start by acknowledging the existence of this thread: Mathematics and cancer research ? It is well-known that mathematical modeling and computational biology are effective tools in cancer ...
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0answers
70 views

Is the group of rational points of an anisotropic absolutely quasi-simple algebraic group over a non-archimedean local field known to be perfect?

Suppose that $G$ is an algebraic group defined over a non-archimedean local field $k$ which is absolutely quasi-simple and anisotropic over $k$. Is it known whether the group $G(k)$ is necessarily ...
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1answer
474 views

euler class of the normal bundle and self intersection number [duplicate]

Let $S$ be a compact submanifold of $X$ smooth manifold. I know that $T_X|_S=T_S\oplus N_{S/X}$ where $N_{S/X}$ is the normal bundle. I have read that the euler class $e(N_{S/X})$ corresponds (via ...
2
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2answers
244 views

Bounds for the milnor number of a hypersurface singularity

I am having a hard time in finding an upper bound in terms of the degree and the dimension for the Milnor number of an isolated hypersurface singularity. I am mostly interested in surfaces on the ...
2
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1answer
168 views

Canonical lifts from $\mathbb F_q$ and CM-theory

One knows that (ordinary) Jacobians of hyperelliptic curves over a finite field $\mathbb F_q$ (mostly of genus 1 (elliptic curves) and 2) are extensively studied by cryptographers, as a platform for ...
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6answers
2k views

Geometric meaning of the Euler sequence on \mathbb{P}^n (Example 8.20.1 in Ch II of Hartshorne)

Is there any geometric way to understand the exact sequence in Example 8.20.1 in Ch II of Hartshorne (p. 182), or its dual from theorem 8.13? Here is the sequence: $0\to O_{\mathbb{P}^n}\to ...
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0answers
236 views

What is the status of the Friedlander-Milnor conjecture today?

For the purposes of this question, the Friedlander-Milnor (FM) conjecture asserts an equality of the group homology for algebraic groups, and their discretizations in the following sense: Conjecture ...
4
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3answers
231 views

Projective submanifolds of $\mathbb CP^n$ whose normals bundles are sums of linear.

Let $X\subset \mathbb CP^n$ be a smooth submanifold whose normal bundle is $$\bigoplus_{i=1}^{codim X}O(k_i).$$ Is there some general enough additional condition of $X$ that implies that $X$ is a ...
4
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1answer
86 views

Hypersurfaces containing a general chain of lines

Let $X$ be a general chain of $d$ lines in $\mathbb P^n$, where $n \geq 3$. Let $I$ be the homogeneous ideal of polynomials vanishing on $X$. What is the Hilbert function $$P(k) = \dim I_k$$ of $X$? ...
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2answers
209 views

Secant Lines contained in Hypersurfaces

If X is a hypersurface of degree d in $\mathbb{P}^n$ and S is the singular locus of X then when is it true that the secant line of two points in S is contained in X? I think this has something to do ...
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2answers
212 views

Rational curved lying in the boundary of Deligne-Mumford compactification $\bar M_g$

Let $\bar M_g$ be the Deligne-Mumford compactifiction of the moduli space of complex genus $g$ curves $M_g$. Is this correct that through every point of the boundary $\bar M_g\setminus M_g$ passes a ...
7
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4answers
534 views

What are the general techniques for proving a variety is not toric?

When I read the paper "A survey on Cox rings" with the link below: http://math.berkeley.edu/~velasco/Survey.pdf In Section 4.2 it is mentioned that the Del Pezzo surface given by blowup of ...
22
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3answers
1k views

What is the algebraic geometry version of the spheres?

In topology the spheres $S^n$ are the "simplest" closed manifolds, and they are like "Dirac's delta at $n$" for (reduced) cohomology groups. Furthermore they are boundaries of the simplest compact ...
3
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1answer
180 views

Is the big cell a principal open set?

Let $G$ be a complex affine reductive algebraic group, $B\subseteq G$ a Borel with maximal torus $T$ and unipotent radical $U$. Let $w\in\operatorname N_G(T)$ be a representative of the longest Weyl ...
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4answers
1k views

Quillen's motivation of higher algebraic K-theory

Almost the same question was already asked on MO Motivation for algebraic K-theory? However, to my taste, the answers there consider the subject from a more modern point of view. When I open a book ...
4
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1answer
200 views

Fermat surface known to have very few rational integer solutions

The motivation for this question is the Selmer curve, given by $$\displaystyle 3x^3 + 4y^3 + 5z^3 = 0.$$ One can show that this curve has no rational integer solutions, despite having a solution ...
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1answer
94 views

Can the property of essential finite type checked at a point?

Let $k$ be a field, and let $A$ be a commutative $k$-algebra which is noetherian. Suppose that for each prime ideal $p$ of $A$, it holds that the field $k(p)$, the field of fractions of $A/p$ has ...
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2answers
761 views

Places to learn about Landau-Ginzburg models

Here is what I know about Landau-Ginzburg models: It is an important player in the story of mirror symmetry. It involves "potentials" which are functions of complex varibles, which have isolated ...
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49 views

Surjectivity of $f\colon M\rightarrow \Gamma_{pR_p}(M_p)$

Let $R$ be a Noetherian ring and let $M$ is finitely generated $R$-module.Suppose $p$ is a minimal prime in $\text{Supp}_RM$. Then $f\colon M\rightarrow \Gamma_{pR_p}(M_p)$ that $f(m)=m /1 $ is ...
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1answer
140 views

Proof of the Belyi's theorem: where it is really used the hypothesis?

Consider the Belyi's theorem: If a smooth projective curve $X$ is defined over $\overline{\mathbb Q}$, then there exists a finite morphism $X\longrightarrow\mathbb P^1(\mathbb C)$ with at most ...
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1answer
73 views

Computing the minimal free resolution of a coherent sheaf on projective space

Most books on commutative algebra explain Grobner bases in the non graded case and minimal free resolutions in the local case. I like projective geometry and want to compute the minimal free ...
12
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2answers
307 views

Do most degree $d$ morphisms of $P^n$ have smooth critical locus?

Let $f=[F,G,H]:\mathbb{P}^2\to\mathbb{P}^2$ be a morphism of degree $d\ge2$. The critical locus $C_f$ of $f$ is the zero-locus of the Jacobian determinant: $$ C_f = \left\{ [x,y,z]\in\mathbb{P}^2 ...
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1answer
195 views

representation of algebraic fundamental group of projective line minus three point

everyone, I want to ask is there any result in the literature similar to the following: Let $ X=\mathbb{P}^1\backslash \{0,1,\infty\}$, then $X$ is defined over $\mathbb{Z}$. Let $X_{\mathbb{Q}}$ ...
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0answers
147 views

Surjectivity of $f\colon\Gamma_Z(M)\rightarrow\bigoplus_{p\in Z\backslash Z'}\Gamma_{pR_p}(M_p)$

Suppose $Z'\subseteq Z\subseteq\text{Spec} R$ such that every element in $Z\backslash Z'$ is a minimal element (with respect to inclusion as ideals) in $Z$. Assume further that both $Z$ and $Z'$ are ...
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61 views

Explicit calculation of module of derivations on noncommutative polynomial ring

Let $R$ be a commutative unital associative ring and set $R<x,y>$ to be the $R$-algebra of non-commuting polynomials in two variables over $R$. Explicitly how would one go about computing ...
3
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1answer
141 views

flat descent for perverse sheaves

Let $E \in D^{b}_{c}(X,\overline{\mathbb{Q}}_{l})$ where $X$ is a $k$ scheme of finite type for a field $k$. Let $Y\rightarrow X$ a finite flat surjective morphism such that $f^{*}E$ is perverse and ...