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

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8
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2answers
407 views

Moduli space of (all) vector bundles on $\mathbb{P}^1$

It is well known that, by a theorem of Grothendieck, every vector bundle (always assumed coherent in this question; and everything is over the complex numbers) splits as a direct sum of line bundles. ...
5
votes
1answer
217 views

Is $MGL$ an $H\mathbb{Z}$-algebra?

Let $\mathrm{MGL}$ be the $\mathbb{P}^1$-ring spectrum over a field $k$ representing algebraic cobordism. Suppose, for simplicity, that $k$ is of characteristic 0. Let $H\mathbb{Z}$ be the motivic ...
-2
votes
0answers
44 views

Explicit equations for morphism determined by a linear system, Part II [migrated]

This is a follow-up to this question: Explicit equation for extension of a rational map? . I realized my wording was a bit unclear and that I could have been more direct in what I was asking for. ...
1
vote
1answer
180 views

Explicit equation for extension of a rational map?

Consider $X = \mathbb{P}^2_k$ and $\mathcal{O}_X(2).$ If we consider the linear system $\mathcal{L}$ of conics passing through the point $[0:0:1]$ we can see that this is spanned by the sections ...
1
vote
1answer
142 views

Local nontriviality of genus-one curves over extensions of degree dividing $6^n$

Suppose $p\geq 5$ is a prime, and $C$ a genus-one curve, defined over $\mathbf{Q}$. Is there always an extension $K/\mathbf{Q}_{p}$ whose degree divides a power of $6$, so that $C(K)$ is not empty? (I ...
1
vote
1answer
633 views

Problems in some parts of Monique Hakim thesis?

In "Reminiscences of Grothendieck and his school" Luc Illusie says: "I heard from Deligne that there were problems in some parts. (of Monique Hakim thesis). Topos annelés et schémas relatifs, ...
1
vote
0answers
68 views

Connectedness of fibers of a map related to the secant variety

Let $X\subsetneq\mathbb{P}^{N}$ be a smooth projective variety, and let $$ S_{X}=\overline{\{(x,y,z)\in X\times X\times \mathbb{P}^{N}:x\neq y, z\in\langle x,y\rangle\}}. $$ The secant variety of $X$ ...
1
vote
0answers
170 views

Introductions to modern algebraic geometry [duplicate]

I was wondering if people have any feelings on the pros and cons of various introductions to algebraic geometry (among the contenders would be Ravi Vakil's notes, Hartshorne, Mumford, Harris, ...
0
votes
0answers
143 views

Chow classes of non-reduced sub-schemes

I am trying to understand the geometric picture of primary ideals, in particular if and how one can define a notion of multiplicity for the underlying geometric 'sets'. My understanding of the subject ...
7
votes
0answers
223 views

Étale morphisms in SDG

On page 70 here, Kock defines a formally étale morphism $f:M\rightarrow N$ as one for which the following square is a pullback for each $d:\mathbf 1\rightarrow D$ $$\require{AMScd} \begin{CD} M^D ...
8
votes
2answers
309 views

The class of the diagonal in the symmetric product of a smooth curve

Let $C$ be a smooth curve of genus $g$, and let us consider its $d$-th symmetric product $\textrm{Sym}^d(C)$ and its Jacobian $J(C)$. Fixing a point $p_0 \in C,$ there are two maps $$u_d\colon C_d \to ...
3
votes
1answer
117 views

is the normalization of a smooth curve in a tamely ramified finite separable extension of the function field also smooth?

Let $X$ be a smooth proper curve over a field $k$, with function field $K$. Let $L$ be a finite separable tamely ramified extension of $K$, and let $Y$ be the normalization of $X$ in $L$. Is ...
8
votes
0answers
228 views

Letters of a bi-rationalist

V.V. Shokurov has written several papers over the course of about 10 years which are called "Letters of a bi-rationalist". Here are the ones that I could find: Letters of a bi-rationalist. I. A ...
2
votes
1answer
103 views

Is torsion submodule of a $p$-adically complete and separated $\mathbb{Z}_{p}$-module closed?

I was asking to myself the following question. Consider a $p$-adically complete and separated topological algebra $R$ over $\mathbb{Z}_{p}$. As $\mathbb{Z}_{p}$ is a domain, we know that the ...
11
votes
2answers
729 views

Gauss proof of fundamental theorem of algebra

My question concerns the argument given by Gauss in his "geometric proof" of the fundamental theorem of Algebra. At one point he says (I am reformulating) : A branch (a component) of any algebraic ...
11
votes
2answers
304 views

Examples of varieties with many automorphisms acting trivially on co-homology

Let $X$ be a smooth projective variety over the complex numbers. Denote by $Aut(X)$ its automorphism group, by $Aut(X)_0$ the connected component of the identity, and by $G$ the quotient ...
7
votes
0answers
196 views

Does Stepanov's method extend to complete intersections?

Stepanov (circa 1970) created the polynomial method to limit the rational points of an algebraic curve over $\mathbb{F}_q$, leading to one of several alternative proofs of Weil's Riemann hypothesis ...
4
votes
0answers
190 views

Reference for Hensel's Lemma in Algebraic Geometry

The following form of Hensel's Lemma in Algebraic Geometry is well-documented in the literature: $\textbf{Theorem 1}$: Let $R$ be an Henselian local ring with maximal ideal $\mathfrak{m}$, and let ...
5
votes
1answer
129 views

Primitive log-divergent graphs and convergence of Feynman amplitudes

To a connected graph $G$, quantum field theory attaches the integral $$ I_G=\int_{\sigma} \frac{\Omega_G}{\Psi_G^2} $$ where $N_G$ is the number of edges of the graph, $\sigma$ is the simplex of ...
3
votes
0answers
239 views

Why is solving polynomial systems NP hard?

Solving polynomial systems is known to be a NP hard problem; however it is not completely clear to me where this complexity comes from. My interest is in the case of systems of multivariate ...
4
votes
0answers
150 views

Descent of line bundles to the quotient

If a finite group acts $G$ on a variety $X$, consider the quotient $X/G$. I would like to understand which line bundle on $X$ descends to $X/G$. The action is not free. Can anyone direct me to some ...
6
votes
1answer
246 views

Smoothness of the “Archimedean special fiber” in Arakelov geometry

If $X$ is a scheme over, let's say, $\mathbb{Z}_p$, one can consider its special fiber obtained by reduction modulo $p$ ans it certainly makes sense to ask if this special fiber is smooth or not. ...
2
votes
1answer
82 views

Are the Prym varieties geometrcally nondegenerate subvarieties of the Jacobians?

A subvariety $V$ of an abelian variety $X$ is geometrically nondegenerate if it meets any subvariety of $X$ of dimension bigger than or equal $codim(V)$. My question is about the Prym varieties as ...
7
votes
1answer
164 views

How fine an invariant of a representation is its quotient singularity?

This is a refinement of a question asked on MSE. Let $G$ be a finite group and let $V$ be a finite-dimensional faithful complex representation of $G$. Consider $V$ as an affine complex variety. In ...
2
votes
1answer
155 views

Families of smooth projective varieties over dvr

Let $R$ be a discrete valuation ring with residue field $k$, an algebraically closed field of characteristic zero and $\pi:X\to \mbox{spec}(R)$ a smooth, projective family of surfaces. Denote by ...
1
vote
0answers
135 views

Quasi-coherator

Recently, I have been studying about quasi-coherator and I have some doubts. 1) I know that quasi-coherator of a sheaf on a scheme exists if the scheme is quasi-compact and semi-separated. Could you ...
1
vote
0answers
78 views

Dimension of a module (which is not necessarily finite)

Let $R=\bigoplus_{ n\in\mathbb N}R_{ n}$ be a Noetherian standard ring defined over an Artinian local ring. Let $M=\bigoplus_{ n\in\mathbb N}M_{ n}$ be an $\mathbb N$-graded $R$-module (not ...
48
votes
2answers
2k views

What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky?

In 1991, Kapranov and Voevodsky published a proof of a now famously false result, roughly saying that the homotopy category of spaces is equivalent to the homotopy category of strict infinity ...
21
votes
1answer
551 views

Useful, non-trivial general theorems about morphisms of schemes

I apologize in advance as this is not a research level question but rather one which could benefit from expert attention but is potentially useful mainly to novice mathematicians. I'm trying to ...
0
votes
1answer
137 views

Hilbert function and numerical polynomial

Let $R=\bigoplus_{ n\in\mathbb N}R_{ n}$ be a Noetherian standard ring defined over an Artinian local ring. Let $M=\bigoplus_{ n\in\mathbb N}M_{ n}$ be an $\mathbb N$-graded $R$-module (not ...
2
votes
0answers
43 views

What is the complexity of finding a distortion map on a supersingular elliptic curve?

Let $E$ be a supersingular elliptic curve which is defined over $\mathbb{F}_q$ and $P\in E$. Then there exist a distortion map with respect to $P$. I am looking for an algorithm which finds the map ...
5
votes
0answers
121 views

The Veronese Surface is the only surface of $\mathbb{P}^{5}$ that can be isomorphically projected to $\mathbb{P}^{4}$

Let $X\subset\mathbb{P}^{5}$ be a surface such that the dimension of its secant variety is $\dim \mathrm{Sec} (X)=4$. In Theorem 3.2.5 of Francesco Russo's book 'Tangents and Secants of Algebraic ...
4
votes
1answer
100 views

Equations for covering spaces of non-singular curves

Suppose I have a Riemann surface $C$ given by an explicit equation or equations as a complex plane curve or space curve. $C$ has a(n unramified) covering space $C'$ associated to each subgroup of ...
1
vote
0answers
41 views

Gauss map of the Veronese embedding of degree 2

This question is related to one I asked previously. Sorry if some of the notation or discussion below is unwieldy or nonstandard. I am still trying to learn the relevant terminology, so it's likely ...
4
votes
1answer
141 views

Deformation long exact sequence of GW theory in the analytical setting

Let $f\!=\!(u\colon (\Sigma,p_1,\ldots,p_k) \to X)$ be an element of the moduli space of genus $g$ $k$-marked degree $A$ $J$-holomorphic maps $\mathcal{M}_{g,k}(X,A,J)$. For simplicity assume ...
21
votes
2answers
500 views

CM $j$-invariants in $p$-adic fields

I'm trying to understand the $p$-adic distribution of $j$-invariants for elliptic curves with complex multiplication. Specifically, suppose $\sigma$ is some embedding $\sigma:\overline{\mathbb Q}\to ...
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
73 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
391 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
216 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 ...
0
votes
0answers
60 views

A representation problem on graphs

Given adjacency matrix $A\in\{0,1\}^{n\times n}$ of a graph denote $A(y,z)$ to be matrix where $0$ is replaced by $z$ and $1$ by $1-z$ on the non-diagonals (replace diagonals by $y$). Denote the ...
1
vote
1answer
145 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 ...