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

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2
votes
1answer
75 views

Torsion free sheaves in flat families

Let $R$ be a dvr, $X$ a flat, projective, integral, normal $R$-scheme such every closed fiber is again integral, normal. Let $F$ be a torsion-free coherent sheaf on $X$, flat over $R$. Is it true that ...
2
votes
0answers
75 views

What is $\mathrm{Num}(X)$ for the canonical cover $X$ of a bielliptic surface $S$?

A bielliptic surface $S$ is a smooth projective complex surface of Kodaira dimension 0 with $h^1(\mathcal O_S)=1$ and $h^2(\mathcal O_S)=0$. It is well known that $S=(A\times B)/G$, where $G$ is a ...
3
votes
0answers
72 views

Sign of bivariate polynomial evaluated over two algebraic numbers

I would like to compute the sign of a bivariate polynomial $f$ evaluated over two algebraic numbers $a$, $b$. The numbers are in "isolating interval representation" meaning that each one is defined by ...
1
vote
0answers
83 views

Chern classes of a resolution of singularities

Let $j:X\subset \mathbb P_{\mathbb C}^n$ ($n\geq 3$) be a hypersurface, defined by a section of a very ample line bundle $\mathcal L$, with a ordinary double point $P$ as the only singularity and ...
12
votes
0answers
375 views

Elkies' supersingularity theorem in higher dimension (in terms of the associated Newton polygon)

Elkies' supersingularity theorem: Given an elliptic curve $E$ over $\mathbb{Q}$, there are infinitely many primes $p$ such that $E$ is supersingular over $\mathbb{F}_p$. I have seen another post on ...
4
votes
0answers
128 views

Confusion with proof of Pieri's Formula

In Manivel's Symmetric Functions, Schubert Polynomials, and Degeneracy Loci, I am confused with some parts of his proof of Pieri's Formula. It is given as Pieri's Formula $3.2.8$ (p. $109$): If ...
8
votes
1answer
224 views

Tube of a mod p point on a smooth Z_(p)-scheme

Let $R$ be a smooth, integral, finite-type $\mathbb{Z}_{(p)}$-algebra of relative dimension $n$ and $\overline{f} \colon R \to \mathbb{F}_p$. Then Hensel's lemma tells us that this lifts to a map $R ...
1
vote
1answer
143 views

Ideal of rational curve in projective space

Let $C_d\subset \mathbb P^d$ be a rational curve of degree $d>2$ ($C_d$ can be reducible) and $n\geq d$. Do we always have $h^0(I_{C_d}(n))=\binom {n+d}{d} - nd-1$?
13
votes
0answers
392 views

Solving polynomial systems with homotopy. Where is the bottleneck?

I have a polynomial system with $n+k$ unknowns ($n+k$ can be greater than 8), that is known to have a limited number of isolated solutions. I want to solve this system numerically, but if I plug it ...
3
votes
1answer
217 views

etale localization reference request

I'm looking for a reference for the following statement: Let $P$ be a property of morphisms of schemes local on the target in the etale topology. Let $f : X\rightarrow Y$ be a morphism of schemes ...
1
vote
1answer
264 views

Local deformations of Lagrangian submanifolds in holomorphic symplectic manifold and their intersections

Let $Y\subset X$ be a Lagrangian submanifold in a holomorphic symplectic manifold $X$. We know that there exists a local moduli space $M$, which parametrizes lagrangian submanifolds in $X$(there are ...
1
vote
1answer
146 views

Hilbert polynomial of the normalization

Let $X$ be a projective non-normal scheme, say over the complex numbers, endowed with an ample line bundle $L$. Let $\nu \colon \hat{X} \to X$ be its normalization. Is it possible that $(X,L)$ and ...
2
votes
2answers
230 views

Curves in homogeneous varieties

Let $C$ be a curve in a projective homogeneous variety $X$. Fixed a general point $x$ in $X$, does there exist a curve $V$ in $X$ passing through $x$ and such that $C$ and $V$ have the same homology ...
2
votes
0answers
101 views

Quotients of quasi affine varieties and extension of scalars

I have some questions about GIT quotients and extensions of scalars of categorical quotients: 1) Let $X$ be a complex algebraic quasi-affine variety, $G$ an algebraic reductive group over ...
1
vote
0answers
90 views

Components of Kontsevich moduli space of stable maps and reducible curves

Let $X\subset \mathbb P^n$ be a smooth projective variety which is Fano and $M_{0,0}(X,e)$ the (projective) Kontsevich moduli space of rational curves of degree $e>1$. Is it possible (or are there ...
18
votes
4answers
793 views

Applications of algebraic geometry to machine learning

I am interested in applications of algebraic geometry to machine learning. I have found some papers and books, mainly by Bernd Sturmfels on algebraic statistics and machine learning. However, all this ...
2
votes
0answers
92 views

A reference about a problem of the number of the rational points on a projective scheme

Let $X\hookrightarrow\mathbb{P}^n_{\mathbb{F}_q}$ be a pure dimensional projective scheme of dimension $d$. So we know a trivial estimate of the number of $X(\mathbb F_q)$ is that $\#X(\mathbb ...
2
votes
1answer
85 views

On a morphism between reflexive sheaves

Let $X$ be a normal, projective variety and $U$ be the regular locus of $X$. Let $\mathcal{F},\mathcal{G}$ be reflexive sheaves on $X$ and $f:\mathcal{F} \to \mathcal{G}$ be a morphism. Suppose that ...
1
vote
1answer
82 views

Components of Kontsevich moduli space of stable maps and multiple covers

Let $X\subset \mathbb P^n$ be a smooth projective variety over $\mathbb C$ which is Fano and $M_{0,0}(X,e)$ the (projective) Kontsevich moduli space of rational cuves of degree $e>1$. Is it ...
8
votes
1answer
296 views

Topology on the space of constructible sheaves

Let $X$ be a nice compact topological space with a fixed finite stratification by locally closed topological manifolds. At the beginning one may assume that $X$ is a complex algebraic manifold with ...
1
vote
0answers
53 views

Polarization of the Prym variety

Let $X\rightarrow Y$ a ramifield double cover of curves, $J_X, J_Y$ their jacobians, $P\subset J_X$ the Prym variety, for any line bundle $L$ on $X$ of degree $g_X-1$, denote by $\Theta_L$ the ...
0
votes
1answer
309 views

How do Modular Forms and the Geodesic Flow interact? [closed]

Textbooks talk at length of the modular properties of $\theta(z)$ or $\tau(z)$ and the prominent role of $SL(2,\mathbb{Z})$ or one of the congruence groups. In that case, aren't the basic objects ...
15
votes
0answers
265 views

Does the “holomorphic spheres-to-continuous spheres” forgetful function respect the mixed Hodge structures on homotopy groups?

For each smooth, projective, complex variety $X$ that is simply connected, John Morgan constructed a natural mixed Hodge structure on the homotopy group $\pi_k(X,x)\otimes \mathbb{Q}$. This was ...
-2
votes
1answer
154 views

Is it easy to see that a cubic surface $V$ in $CP^3$ has no holomorphic 2-forms? [closed]

More specifically, what facts do you need to know to conclude $H^2(V) = H^{1,1}$? In general, are there hypersurfaces in $CP^n$ without holomorphic $k$-forms for some $k$?
12
votes
1answer
278 views

Independence of $\ell$ of Betti numbers

When $X$ is a smooth proper variety over $\mathbb F_q$, we know by Deligne's theory of weights that the dimension of $H^i_{\operatorname{\acute et}}(\bar X, \mathbb Q_\ell)$ does not depend on $\ell$. ...
28
votes
1answer
2k views

Grothendieck says: points are not mere points, but carry Galois group actions

Apologies in advance if this question is too elementary for MO. I didn't find an explanation of these ideas in any algebraic geometry books (I don't know French). The following is an excerpt from ...
6
votes
1answer
306 views

Quantum Grassmannians?

In noncommutative algebraic geometry a commonly studied family of objects are quantum projective spaces. Theses are certain deformations of the homogeneous coordinate ring of $\mathbb{CP}^n$. For ...
4
votes
0answers
229 views

Narasimhan-Simha Hermitian metric vs Weil-Petersson metric

What is relation between Weil-Petersson metric on holomorphic fibre space $f:X\to Y$ of compact complex manifolds $X,Y$ . (let fibres are Calabi-Yau manifolds) And Ricci curvature of Narasimhan-Simha ...
2
votes
3answers
180 views

Isolating roots of polynomial system

I would like to isolate the regions which contain the roots of a system of two bivariate cubic polynomials. I thought I would project the solutions onto $x$ and $y$ axis by means of resultant ...
2
votes
1answer
76 views

Grassmannian inside a hyperkahler manifold

I am currently looking at stratified Mukai flop. Roughly speaking, this is a construction that, starting with a grassmannian $G$ inside a hyperkahler $X$ produces a birational manifold $X^*$ (with a ...
4
votes
0answers
97 views

relation between constructible function on quiver variety and representation of Kac Moody algebra

This is a follow-up question of my previous question. In Lusztig's paper Section 12, Lusztig constructed an algebra embedding from $U\frak{n}_-$ to $\oplus_V M(\Lambda_V)$. In Nakajima's paper ...
2
votes
0answers
82 views

Basic facts about the join of two varieties

Let $Y\subseteq X$ be a smooth subvariety of the smooth projective variety $X\subseteq\mathbb{P}^{N}$. Let us consider $$ S_{X,Y}:=\overline{\{(x,y,z)\in X\times Y\times \mathbb{P}^{N}:x\neq y, ...
2
votes
0answers
124 views

Birational map of surfaces with target nonsingular

There is a question in a certain well-known textbook that goes like this. Let $f:X'\rightarrow X$ be a birational morphism of surfaces (integral separated schemes of dimension two and of ...
23
votes
2answers
795 views

Steenrod operations in etale cohomology?

For $X$ a topological space, from the short exact sequence $$ 0 \rightarrow \mathbb{Z}/2 \rightarrow \mathbb{Z}/4 \rightarrow \mathbb{Z}/2 \rightarrow 0 $$ we get a Bockstein homomorphism ...
3
votes
2answers
236 views

Borel--Bott--Weil for the Grassmannians

The Borel--Bott--Weil Theorem is usually stated for the complete flag manifold of $SU(N)$. Does an analogue hold for the other flags, for example the Grassmannians? More precisely, suppose $G(\mathbf ...
5
votes
1answer
134 views

Symmetric polynomial separating points

I've been looking for references/answers to this problem for several days and I couldn't find anything. If we consider the closed unit ball $B$ in $\mathbb C^2$ then for any point $(z_1,z_2)\notin B$ ...
20
votes
3answers
736 views

Does X(13) have potentially good reduction at 13?

The complete level modular curve $X(p)$ does not have potentially good reduction at $p$ for any $p \neq 2,3,5,7,13$ because then there are cusp forms on $X_0(p)$ showing up in the cohomology of ...
0
votes
1answer
199 views

Is it true that singular fibers of elliptic fibrations that have the same Kodaira type are isomorphic schemes?

It is well known that Kodaira gave an essentially topological classification of the possible singular fibers of elliptic fibrations according to their type: ...
3
votes
0answers
101 views

groupoids representing mapping stacks

1)Let $X$ be a differentiable stack ((2,1) sheaf over the category of smooth manifolds $Man$) and that is geometric. Let $N\in Man$, then $$ Map(y(N),X) $$ is again a differentiable stack ($y$ is the ...
3
votes
0answers
91 views

Moduli space of null Sasaki $η$-Einstein structures for higher dimensions(Calabi-Yau structures in Sasakian setting)

The moduli space of null Sasaki $η$-Einstein structures for simply connected compact 5-dimensional manifold $M$ is determined by the following quadric $$\{[\alpha]\in H^2(M,\mathbb C) \; \text{such ...
2
votes
0answers
131 views

Question regarding a lemma in Principles of Algebraic Geometry

My question is regarding the lemma on page 81 of the book by Griffiths and Harris. The lemma says the following: A $\bar{\partial}$-closed form $\psi\in Z^{p,q}_{\bar{\partial}}(M)$ is of minimal ...
3
votes
0answers
112 views

Automorphism groups of elliptic bundles

This is a question in complex geometry, but even for algebraic varieties I don't know the answer: Let $S$ be a smooth compact Kähler surface (for example a smooth complex projective surface) that is ...
1
vote
0answers
67 views

Roots of generalized homogeneous polynomials

A polynomial $P(\xi)$ of $n$ real or complex variables is homogeneous of order $m$ with respect to $\lambda \in \mathbb{Z}_{+}^n$ if $$P(\xi) = \sum\limits_{\substack{(\alpha, \lambda) = m \\ \alpha ...
3
votes
0answers
113 views

Flatness of Weil restriction

Let $X\rightarrow Y$ a ramified double cover of smooth projective curves, and let $$\mathcal G:=Res_{X/Y}(SL_n)$$ be the Weil restriction of the constant group scheme $SL_n$ over $X$. Question: ...
18
votes
1answer
636 views

How to stop worrying about enriched categories?

Recently I realized that ordinary category theory is not a suitable language for a big portion of the math I'm having a hard time with these days. One thing in common to all my examples is that they ...
3
votes
1answer
139 views

Identifying the canonical principal polarization of a Jacobian

Let $X$ be a curve over an algebraically closed field $k$ (even over $k = \mathbb{C}$ if you want), let $J = Pic^0_{X/k}$ be its Jacobian, let $P \in X(k)$ be a point, and let $i \colon X ...
2
votes
1answer
104 views

Uniqueness of a (weighted) affine cone

Let $Z$ be a projective variety embedded into $\mathbb P^n$. Then we can define an affine cone over $Z$ as the inverse image of $Z$ under canonical map $\mathbb A^{n+1}\setminus0 \to \mathbb P^n$. I ...
2
votes
0answers
107 views

Is the normalization of a blowup of a LC variety again LC ?

If I blow up a subvariety Z in a log canonical variety X and then normalize, is the resulting variety X' log canonical? I reason as follows. Let Y-->X' be a strong resolution of singularities. If ...
4
votes
0answers
107 views

Littlewood-Richardson rule for the complete flag variety: GapP complete?

The cohomology ring of a complete flag variety $X$ has a basis of Schubert classes $S_u$ for permutations $u$. Define the Littlewood-Richardson coefficient $c_{uv}^w$ for permutations $u,v,w$ to be ...
4
votes
0answers
248 views

Algebraic curves that enclose and exclude given points in the plane

Q1. Given two finite sets $R,G$ of points in $\mathbb{R}^2$, $|R|=r$ red points and $|G|=g$ green points, is it always possible to find a simple closed algebraic curve $C(x,y)=0$ that ...