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

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Supplementary notes to Mumford's The Red Book of Varieties and Schemes

I am a graduate student with good mathematical maturity (I took advanced courses like category theory, commutative algebra...). I want to study algebraic geometry from Mumford's red book. I find it ...
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23 views

Irreducible component of a scheme over a dvr

Let $\mathcal M$ be a (reduced) quasi-projective scheme over a dvr (of mixed caracteristics), $R$. Suppose that the generic fiber $\mathcal M_{\eta_R}$ is (nonempty) smooth and irreducible of ...
3
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1answer
90 views

Polynomials with Unique Critical Value

My question is extremely simple to state: I am looking for a characterization of multivariate complex polynomials $f$ such that $f(Sing(f))=\{0\}$. My motivation is that I recently read somewhere that ...
2
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1answer
101 views

Fibrations on blow-ups of $\mathbb{P}^2$

Let $X_n = Bl_{p_1,...,p_n}\mathbb{P}^2$ be the blow-up of $\mathbb{P}^2$ in $n$ general points $p_1,...,p_n\in\mathbb{P}^2$. Let $f_i:\mathbb{P}^{2}\dashrightarrow\mathbb{P}^1$ be the linear ...
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0answers
46 views

Set of smooth curves on the Hilbert scheme is open. H

Let $H = Hilb_{d,g,r}$ be the Hilbert scheme of genus $g$ curves of degree $d$ in proyective space $\mathbb{P}^r$. Is it true that the set of points of $H$ consisting of smooth curves is open in ...
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24 views

Uniqueness of Riemann Constant Vector Solution

Let $X$ be a compact, genus $g$ Riemann surface (given as the desingularization and compactification of a plane algebraic curve), $J(X)$ its Jacobian, and $A : X \to J(X)$ the Abel map $$A(P) = ...
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95 views

Bundles over Grassmanian with given top Chern class

So, I have been working on Chern classes for my master thesis and apparently (My proofs could be wrong and a few things are still vague) I was able to give a construction method and exhibit, via ...
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37 views

Calculations about the normal bundle of embedding of symmetric products

Suppose $C^{(d)}$ is the $d$-th symmetric product of a curve, embed it in $C^{(d+s)}$ by $E\to E+sP_0$ where $P_0$ is a fixed point. The normal bundle of this embedding is denoted by $N$. Suppose ...
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78 views

Universal property of complete linear systems

Let $X$ be a projective scheme over a field $k$ and $S$ a $k$-scheme. Fix a closed immersion $i:X \to \mathbb{P}^n$ for some $n$ and denote by $\mathcal{O}_X(1):=i^*\mathcal{O}_{\mathbb{P}^n}(1)$. Let ...
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109 views

Flat cohomology of an ordinary liftable Calabi-Yau threefold

Let $k$ be a perfect field of characteristic $p>0$ and consider an ordinary liftable Calabi-Yau threefold $X_{0}/k$. By this I mean that $H^{i}(X_{0},B_{X_{0}/k}^{j})=0$ for all $i\geq 0$ and ...
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249 views

Do modular forms show up in the cohomology of moduli spaces of unmarked curves?

Let $\overline{\mathcal M}_{g,n}$ be the compactified Deligne-Mumford moduli stack (although I don't think taking the coarse moduli space will make much of a difference here). If we decompose $g = 1 + ...
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1answer
118 views

Stacks with representable morphisms to algebraic stacks

If $Y$ is an algebraic stack over a scheme $S$ and $X$ is a stack such that there exists an $S$-morphism $X\to Y$ representable by algebraic spaces, then is $X$ an algebraic stack (in the sense that ...
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1answer
359 views

Pure motives and compatible systems of $\ell$-adic representations

I am trying to understand the statement of the conjectures of Deligne on special values of certain $L$-functions, from his article titled, "Valuers de Fonctions L et periodes d'integrales" which ...
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102 views

Is any blow-up of smooth subvarieties always an extremal contraction?

Let $X$ be a smooth complex projective variety and $Z$ be a smooth subvariety of $X$. Take the blow-up $\pi: Y \to X$ of $X$ along $Z$. Then I want to know whether $\pi$ is the contraction of an ...
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110 views

Existence of a map between curves

Given two algebraic curves defined over the rationals, is there a method for determining whether there exists a surjective map from one curve to the other? For instance, suppose X and Y are affine ...
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0answers
58 views

Motivic Pfister type varieties and norm varieties

Due to results of Rost it is known that the Grothendieck-Chow motiv of a Pfister quadric $X$ belonging to a pure $\alpha \in H^n(k,\mu_2)$ is decomposable in the following way $M(X) = ...
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2answers
366 views

Relationship between étale and topological $K(\pi,1)$s

I was trying to find a proof, or a counterexample to the claim that if $X/\mathbb{C}$ is connected smooth projective, then $X$ is a $K(\pi^{\mathrm{\acute{e}t}},1)$ if and only if $X^\mathrm{an}$ is a ...
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113 views

Is the complex structure on a del-Pezzo surface a regular complex structure?

Let $(X, \omega, J)$ be a compact symplectic manifold with an almost complex structure. Fix some homology class $\beta \in H_2(X, \mathbb{Z})$. An almost complex structure $J$ is said to be ...
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0answers
47 views

Geometric interpretation of cubic curve? [migrated]

Lines and conics have clear geometric meanings that are coordinate-free, but cubics seem to rely entirely on cubic equations and coordinate systems. Are there ways to define cubic curves without cubic ...
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1answer
90 views

section of reduced structure map

Let $R$ be a commutative ring whose characteristic is either prime or $0$, such that $R/N$ is an integral domain, where $N$ is the nilradical, and $p: R \rightarrow R/N$ the canonical map. Is there a ...
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57 views

The Zariski closure of subset connected [on hold]

Let $R$ be a commutative ring and $\{P_i\}_{i\in I}$ be an arbitrary subset of $Spec(R)$ (where $Spec(R)$ is the set of all prime ideals of $R$) such that $\dfrac{R}{\bigcap_\limits{i\in I}{P_i}}$ is ...
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1answer
277 views

Automorphisms of finite order in $Out(\widehat{F_2})$

Let $\widehat{F_2}$ be the pro-$\ell$ completion of the free group of rank 2, where $\ell$ is some prime. Every outer automorphism of $F_2$ induces an outer automorphism of $\widehat{F_2}$, hence an ...
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2answers
166 views

Invariants of the maximal unipotent subgroup of GL(n) acting on the space of n by n matrices

Let $G=GL(n,\mathbb{C})$ and let $U\subset G$ be a maximal unipotent subgroup. (For example,assume that U is the set of upper triangular matrices with ones in the diagonal.) Now let ...
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1answer
160 views

Bézout's theorem for arcs in the plane

Consider two polynomials $p,q \in {\mathbb R}[x,y]$, both of degree $d$. Let $\gamma_p$ and $\gamma_q$ be the two curves in ${\mathbb R}^2$ that are defined by these polynomials, and assume that these ...
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3answers
1k views

Intuition behind the Kodaira Vanishing Theorem?

As the question suggests, what is the intuition behind the Kodaira Vanishing Theorem? The Kodaira Vanishing Theorem says that the cohomology groups $H^q(M, L \otimes K_M)$ vanish for $q \ge 1$ when ...
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1answer
231 views

The scheme $y^n = x^{2n}$ for $n$ a rational number [closed]

Let $n\geq 1$ be an integer. If $A$ is a ring, then the spectrum of $A[x,y]/(y^n - x^{2n})$ is a well-defined (affine) scheme, say $X_n$. This scheme describes the "variety" given by the equation ...
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73 views

I want an example of a monomial curve [closed]

Example of a monomial curve and its defining ideal. I am looking for precise definition of a monomial curve and its defining ideal. If possible please tell the minimal generator of this ideal
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1answer
348 views

What is the normal closure of $GL_2(\mathbb{Z})$ inside $GL_2(\mathbb{Z}_\ell)$?

This weird problem popped up in my research: Let $\ell$ be a prime. Is there a description of the smallest normal subgroup of $GL_2(\mathbb{Z}_\ell)$ containing $GL_2(\mathbb{Z})$? Is there a ...
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56 views

local universal sheaf (moduli of stable sheaves)

I do not know much about moduli of sheaves and I wanted to shows that for a smooth (projective) family over a discrete valuation ring of mixed characteristics (relative dimension 3), the locally free ...
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122 views

“Generalized theory of polynomials” for a given commutative Lawvere Theory

I am trying to understand Nikolai Durov's "New Approach to Arakelov Geometry" right now and it got me thinking about a particular thing. Let $R$ be a commutative, associative ring with unit. We can ...
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99 views

Question about Castelnuovo-Mumford regularity

let $R$ be a Noetherian ring an $I$ an ideal of $R$. If $n,m\in N$ and $reg(G(I))=n$, then what can we say about $reg(G(I^m))$? Here $G(I)$ is the associated graded ring.
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179 views

Can view the connected component of the Picard scheme $\text{Pic}^0(X)$ as a “kernel” of the first Chern class?

So on a curve, $\text{Pic}^0(X)$ is just the Jacobian variety, and just correspond to degree $0$ divisors. One way to extend the notion of divisors corresponding to a vector bundle is taking the first ...
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1answer
174 views

pull back of an ample line bundle under a blow up

Suppose $\mu:X\rightarrow Y$ is a blow up of a smooth irreducible subvariety $Z$ of $Y$. Let $L$ be an ample line bundle on $Y$. Let $E$ be the exceptional divisor of $f$. Is it true that there ...
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1answer
162 views

Automorphisms of del Pezzo surfaces

Let $S$ be a del Pezzo surface of degree six over $\mathbb{C}$. Then $S$ is the blow-up of $\mathbb{P}^2$ in three general points $p_1,p_2,p_3$. Is it true that its automorphism group is ...
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81 views

When is a homomorphic image of a polynomial ring self injective [closed]

Following Utumi (but adjusting the definitions for commutative rings): A commutative ring with a unit element is self injective if it is injective as a module over itself. A commutative ring $R$ with ...
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1answer
74 views

Irreducible components of a cone

Suppose $B=A\oplus S^1\oplus S^2\dots$ is a graded ring, $B$ is generated by $S^1$, $C=\textrm{Spec}B$ is called a cone over $X=\textrm{Spec}A$. We have natural projection $\pi\colon C\to X$. ...
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1answer
100 views

Example of indecomposable self injective ring

Is there any example of an indecomposable self-injective commutativr ring with 4 or more maximal ideals?
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1answer
550 views

Remark 4.23.4 in Hartshorne

Crosspost from math.stackexchange, since it's quite possible I might not get a response there. Remark 4.23.4 in Chapter IV of Hartshorne's Algebraic Geometry references a paper by Elkies that ...
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2answers
281 views

Field of definition of dominant morphisms

Let $k$ be an algebraically closed field and $k_0$ a sub-field. Let $X,Y$ be two projective varieties defined over $k_0$. Suppose that that there exists a dominant morphism $f$ between $X_k=X\otimes ...
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0answers
94 views

Stratification of Grassmanian by coordinate hyperplanes

I'm interested in what is known about the stratification of the Grassmanian induced by the coordinate hyperplanes in the Plucker embedding. More specifically, if we view the projective space as a ...
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0answers
364 views

Are there connections between Homotopy type theory and Grothendieck's theory of motives? [closed]

Are there any "visions" (maybe "dreams"), future plans or connections between Homotopy type theory and Grothendieck's theory of motives (or at least "connections" with universal cohomology theory)?
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635 views
+300

A question on representation of graphs

Take a complete graph $K_n$. You want to assign a vectors from $\Bbb F_2^d$ to every edge such that sum of vectors in every simple cycle does not sum to $0$ vector. The question is what is minimum $d$ ...
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104 views

Hasse diagrams of G/P_1 and G/P_2

in the Paper http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.30.5052&rep=rep1&type=pdf at the end, we can see Hasse diagrams for several projective, homogeneous $G$-varieties for $G$ ...
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1answer
220 views

Torsors trivializing over a fixed finite etale cover

Let $S$ be an integral regular scheme and let $T\to S$ be a finite etale morphism. Let $G$ be a smooth affine finite type group scheme over $S$. Is the set of $S$-isomorphism classes of $G$-torsors ...
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1answer
528 views

How to compute this $\mathrm{Ext}^1$?

Let $A$ be a regular local $\mathbb{C}$-algebra of dimension $2$, such as the localization of $\mathbb{C}[x,y]$ at $(x,y)$, and let $\nu=(\nu_1\geq\nu_2\geq\cdots\geq\nu_{\ell}\geq0)$, ...
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1answer
174 views

Lifting torsors in characteristic $p$ to characteristic zero

Let $R$ be a local integral domain with residue field $k$ such that $R$ is of characteristic zero and $k$ is of characteristic $p>0$. Let $G$ be a smooth finite type affine group scheme with ...
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1answer
204 views

Limits of complex projective varieties in the Hausdorff topology on closed subsets of CP^n

In this question a variety means any subset of complex projective space $\mathbb CP^n$ that is the set of common zeroes of a set of homogeneous polynomials. Thus if $A,B\subset{\mathbb C}P^n$ are ...
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182 views

Unirationality over $\mathbb{Q}$

It is known that all smooth projective quartic hypersurfaces of suitably large dimension are unirational over $\overline{\mathbb{Q}}$. Are there any results regarding unirationality over $\mathbb{Q}$ ...
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1k views

Euler characteristic of a curve

Is it a complete coincidence that the Euler characteristic of a curve is exactly twice the Euler characteristic of its structure sheaf?
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165 views

Is there a well-known tautological bundle over $\mathbb{P}(\Lambda^nV)\times V$?

Let $V$ be a vector space of dimension $>n$, and define the subset $$ K:=\{ ([\omega],v)\mid v\wedge\omega=0 \}\subset\mathbb{P}(\Lambda^nV)\times V\, . $$ Denote also by $\pi:K\longrightarrow ...