Questions tagged [ag.algebraic-geometry]
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
21,611
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Examples of curves $C$ with $\operatorname{Jac}(C) \cong E^3$, $E$ a CM elliptic curve
Let $k$ be a field of your choice— I'm particularly interested in algebraically closed fields. Are there explicit examples of curves over $k$ whose Jacobian is isogenous to the product of three copies ...
- 289
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55
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The equation of cubic surface
I was studying the nature of singularities of cubic hypersurface on $\Bbb P^{n+1}$. The chosen form was
$$f(x_0,x_1 \dots , x_{n+1})=x_0^3+x_1^3+\dotsb+x_{n+1}^3-c(x_0+x_1+\dotsb+x_{n+1} )^3=0.$$
I ...
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51
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Sheaves which are locally free on every subscheme of dimension zero
Let $\mathscr{F}$ be a coherent sheaf on a scheme $X$ with reasonable assumptions.
Obviously if I restrict to any point $x \in X$, the restriction $\mathscr{F}|_x$ is free over $x$.
I am interested in ...
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95
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Finiteness of the Brauer group for a one-dimensional scheme that is proper over $\mathrm{Spec}(\mathbb{Z})$
Let $X$ be a scheme with $\dim(X)=1$ that is also proper over $\mathrm{Spec}(\mathbb{Z})$. In Milne's Etale Cohomology, he states that the finiteness of the Brauer group $\mathrm{Br}(X)$ follows from ...
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"Noetherianess" of $\mathrm{Mod}(\mathbb{F}_{p,\square})$
In classical commutative ring theory it is quite immediate to see, that a field is noetherian in the following sense:
For any finitely generated $k$-vectorspace $M$, any sub-object is finitely ...
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85
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Monomorphism which is locally of finite presentation
$\DeclareMathOperator\Spec{Spec}$Let $X$ be an affine scheme of finite type over a field. Let $Y\to X$ be a monomorphism of schemes, which is locally of finite presentation. Is it true that $f$ is ...
- 11
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Artin's "Autoduality of the Jacobian"
In some of his papers (for example, in "Formal groups arising from algebraic varieties" with B. Mazur), M. Artin cites
M. Artin and B. Wyman, Autoduality of the Jacobian, Bowdoin College, ...
- 995
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Infinitesimal neighborhood and Ext group
$\DeclareMathOperator\Ext{Ext}$Let $X$ be a smooth projective complex variety and $\iota\colon Y\subset X$ be a smooth closed subvariety. It is well-known that there is a spectral sequence
$$E_2^{p,q}=...
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218
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Endomorphism ring of a generic elliptic curves in positive characteristic
Let E be a generic elliptic curve over an algebraically closed field $k$ of characteristic $p>0$ (i.e. an elliptic curve corresponding to a geometric point over the generic point of $M_{1,1}$).
...
- 177
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Two different resolution of a three fold
Let $X\subset \mathbb{A}^4_\mathbb{C}$ be a three fold defined by the equation $xy-zw=0$
This variety has a singularity at origin of $\mathbb{A}^4_\mathbb{C}$
If we blow up this three fold in two ways ...
- 83
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How to decompose a given polynomial by ideal generators
Given a finite set of polynomials $f_1, f_2,..., f_n$ of variables $x_1,...,x_m$, generating the ideal $I$, suppose that we have one more polynomial $g\in I$.
What is the algorythm for decomposing $g$ ...
- 621
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67
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Is inverse image along finite group quotient $t$-exact for the perverse $t$-structure?
Let $q: \mathbb{C}\to \mathbb{C}$ be the quotient by $\mathbb{Z}/n\mathbb{Z},$ i.e. the map taking $z\mapsto z^n$. In the accepter answer to Operations on perverse sheaves on disk the inverse image of ...
- 1,021
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A pushout diagram of derived categories coming from an open cover of schemes
Suppose $X=U\cup V$ is the standard open cover of $X=\mathbb{P}^1$ by two affine lines. The descent theorems say that the diagram (with all arrows restriction maps)
$\require{AMScd}$
\begin{CD}
D(X) @&...
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Reference for packing property and König property
Can someone please suggest reference material to study about the packing property and König property of ideals and some examples?
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What are the relations among canonical basis and dual canonical basis of the group of $A_4$? [closed]
How to construct the dual canonical basis of the $A_4$ type group from its canonical basis?
or What are the relations among canonical basis and dual canonical basis of the the $A_4$ type group?
Thank ...
- 1
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2
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587
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A version of Hilbert's Nullstellensatz for real zeros
$\newcommand\R{\Bbb R}$Let $Q(x_1,\dots,x_n)\in\R[x_1,\dots,x_n]$ be an irreducible polynomial such that the dimension of the set $Z:=\{(x_1,\dots,x_n)\in\R^n\colon Q(x_1,\dots,x_n)=0\}$ (defined, say,...
- 118k
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Line of sight calculation - human eye [closed]
background: there is a plot of land that has a high point looking over toward the ocean/horizon downhill. A large bank of trees forms 100 feet away from the hill down toward the ocean blocking the ...
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183
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Atiyah sequence of a coherent sheaf
I want to show that if $X$ is a smooth complex projective variety, then analytification induces an equivalence of categories between algebraic integrable connections on $X$ and analytic integrable ...
- 361
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On the structure of the zeros of real polynomials of several real variables
Let $P(x_1,x_2,...,x_n)$ be a polynomial with real coefficients in the real variables $x_1,x_2,...,x_n$ that vanish on the real quadratic surface $Q(x_1,x_2,...,x_n )=0$ where $x_1,x_2,...,x_n$ are ...
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Leibniz formula for Fulton's divided difference operators for quantum Schubert polynomials
Schubert polynomials are polynomials in the ring $\mathbb{Z}[x]$ where $x$ is an infinite set of variables indexed by the positive integers and they can be expressed in terms of "standard ...
- 2,038
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Vanishing of chow group of 0-cycles for affine, simplicial toric varieties
Let $k$ be an algebraically closed field of characteristic zero.
Let $X$ be an affine, simplicial toric variety over $k$.
If $X$ has dimension one, then it is the affine line over the field $k$, so ...
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66
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When does sum of algebraically independent polynomial become dependent?
Given $f_1,...,f_n \in \mathbb{F}[x_1,...,x_n]$ where $f_n = g + h$. Suppose the sets $\{ f_1,...,f_{n-1},g \}$ and $\{ f_1,...,f_{n-1},h \}$ are algebraically independent then is there a ...
- 159
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Some naive questions about pro-etale cohomology
Here's my dilemma: I'm trying to do some work with the pro-étale topology and would like to compute very basic instances of pro-étale cohomology to see what I can extract from a pro-étale Kummer exact ...
- 472
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56
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Geometric intuition behind hyper-sphere volume recurrence relation [closed]
There is a recurrence relation for calculating the volume of a hyper-sphere and a logical explanation for why it holds, as well as other explanations here on MO. Is there a geometric intuition behind ...
- 1
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93
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Direct image of a vector bundle defined on an open subscheme
Let $X$ be a noetherian irreducible scheme, of dimension $\geq 2$, $P\in X$ a closed point and $U=X\setminus\{P\}$. Let $\iota:U\to X$ be the inclusion morphism.
Let $E$ be a vector bundle on $U$, and ...
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2
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Are Chern classes always vertical?
Let $c_k \in H^{2k}(M, \mathbb{Z})$ be the $k$-th Chern class of the tangent bundle of a Hermitian manifold $M$.
Is $c_k$ necessarily vertical, i.e.
$$
c_k = \sum_{i_1,\dots, i_{k}} \alpha_{i_1 \dots ...
- 105
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217
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On zeros of real polynomials in two variables
Let $P(x,y)$ be a polynomial with real coefficients in two real variables $x,y$ such that the set of zeros of $P(x,y)$ is the real conic curve $Q(x,y)=0$. Will it be true that there exists a ...
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Representability of moduli problem of elliptic curves with complex multiplication
I'd like to know whether the moduli problem for elliptic curves with complex multiplication by a fixed imaginary quadratic number field $K$ (and with suitable level structure to be picked) is ...
- 79
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253
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Gluing together the moduli stacks of elliptic curves over Z[1/2] and Z[1/3]?
I have a long-running desire to understand what is the "global" moduli stack of elliptic curves, as a stack over $\mathrm{Spec}(\mathbb{Z})$.
Recently I was pointed to Katz and Mazur's book, ...
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82
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Smooth non-complete intersection in $(\mathbb{C}^*)^n$?
Are there examples of smooth irreducible subvarieties of $(\mathbb{C}^*)^n$ of dimension $d$ that cannot be cut out scheme theoretically by $n-d$ Laurent polynomials? If yes, how to construct them? ...
- 2,883
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References for orbifold curves
I am looking for a good reference (if there is any) for the theory of orbifold curves from the perspective of stacks. By an orbifold curve I mean something like a $1$-dimensional irreducible Deligne-...
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Is it decidable whether a statement about reals (in the language of ordered rings) is constructively provable?
The language of ordered rings is a first-order language with operators for $+$, $-$, and $\cdot$, constants for $0$ and $1$, and relations for $<$, $=$ and $>$.
To decide whether such a ...
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What is the correct definition of intermediate Jacobian for this singular threefold?
I am considering blow up of $\mathcal{C}\subset(\mathbb{P}^1)^3$, $X=\operatorname{Bl}_{\mathcal{C}}(\mathbb{P}^1)^3$, where $\mathcal{C}$ is a curve given by $$\{s^2u=0\}\subset\mathbb{P}^1_{s:t}\...
- 1,952
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50
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Parabolic (double) quantum Schubert polynomials Pieri formula
I am writing calculation software for computing structure constants of equivariant quantum Schubert polynomials and I discovered that partial flag varieties corresponding to parabolic subgroups have ...
- 2,038
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Hodge bundles associated to a family of complex manifolds
I'm reading Voisin's books on Hodge theory. In the first volume she claimed but didn't prove this theorem:
Theorem 10.10 (Voisin) Let $\varphi:\chi\rightarrow B$ be a family of compact complex ...
- 21
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Pullback morphism of a hyperplane inclusion is zero in the derived category
Let $L \subset \mathbb{C}^n$ be a hyperplane and let $i:L \to \mathbb{C}^n$ be the inclusion. Since $i$ is proper, we have induced maps $i^*: H^k_c(\mathbb{C}^n) \to H^k_c(L)$, and these maps are zero ...
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1
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106
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Continuous extensions of tangent vector fields
Let $\Omega$ be an open subset of $S^2$ with $\bar{\Omega}\neq S^2$. Suppose a continuous tangent vector field $G$ is given on $\partial \Omega$ with $|G(y)|=1$ for all $y\in \partial \Omega$. Does ...
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Canonical comparison between $\infty$ and ordinary derived categories
This question is a follow-up to a previous question I asked.
If $\mathcal{D}(\mathsf{A})$ is the derived $\infty$-category of an (ordinary) abelian category $\mathsf{A},$ then the homotopy category $h\...
- 1,119
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125
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GIT quotient and orbifolds
Let $G$ be a connected complex reductive group. Suppose $G$ acts on a smooth complex affine variety $X$. Assume the stabiliser $G_x$ of every point $x\in X$ is finite. Is it true that $X/\!/G$ is an ...
- 2,681
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220
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Compactification of rigid-analytic varieties
Is it true that any separated quasi-compact rigid-analytic variety embeds into a proper one?
For my purpose, the base field is a $p$-adic number field.
I have seen Huber's universal compactification ...
- 91
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Compact complex manifolds with nef canonical bundle have nonnegative Kodaira dimension
Let $X$ be a compact Kähler manifold with nef canonical bundle. The (Kähler extension of the) abundance conjecture asserts that $K_X$ is semi-ample, and thus $K_X^{\otimes m}$ admits a section for ...
- 265
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151
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Blowup formula for a morphism
Let $f: X\to S$ be a smooth projective morphism between smooth schemes over $\mathbb C$, $i: Z \to X$ a closed subscheme of codimension $c$, also smooth over $S$, and let $g: Y\to S$ be the blowup ...
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Is pullback map on sheaf cohomology injective for surjective morphisms?
Consider a surjective map $f\colon X\to Y$ of smooth projective varieties. It is well known (see e.g. Voisin's Hodge theory I, Lemma 7.28) that the map $H^i(Y,\mathbb Q)\to H^i(X,\mathbb Q)$ is ...
- 2,235
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Solution formula in an explicit equation over $\mathbb{F}_p^3$
I'm looking into a formula involving prime numbers $p \geq 7$ and an equation's solutions. The equation in question is:
$$z^2 = (x^2 - 4x)(y^2 - 4y)((x + 1 - y)^2 - 4x),$$
where $(x,y,z)\in \mathbb{F}...
- 71
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Homotopy coherence datum for composition of Becker-Gottlieb transfers
I have a question about certain detail in following answer by Denis Nardin adressing the concept of presheaves with transfer (mostly known in constructions in motivic homotopy theory) from viewpoint ...
- 6,000
12
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500
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Injective ring homomorphism from $\mathbb{Z}_p[[x,y]]$ to $\mathbb{Z}_p[[x]]$
Is there an injective $\mathbb{Z}_p$-ring homomorphism from $\mathbb{Z}_p[[x,y]]$ to $\mathbb{Z}_p[[x]]$?
- 1,651
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858
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What is this huge generalization of the Modularity Theorem?
A friend of mine wrote:
The point is of course that the Modularity Theorem (as I stated it) is/should be really just a special case of some much bigger theorem which sets up a bijection between ...
- 21.5k
3
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193
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Spectrum of ring in algebraic geometry vs spectrum of Banach algebra
For a commutative unital Banach algebra $A,$ and $x\in A,$ we have $\lambda \in \sigma_A(x)$ if and only if $\phi(x) = \lambda$ for some algebra homomorphism $\phi:A \to \mathbb C.$ The set of all ...
- 1,641
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85
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Vandermonde-type factorization of moment matrix?
Consider $n,d \in \mathbb{N}_{>0}$, there are many functions $y:\mathbb{N}^{n} \to \mathbb{R}$. Now for simplicity, we denote $y(\alpha)$ to be $y_{\alpha}$. Let $|\alpha| = \sum_{i=1}^{n}\alpha_{i}...
- 195
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Is it true that monomorphisms of local Artinian $\mathbb{R}$-algebras are regular?
A Weil algebra is a finite-dimensional real algebra, in which each element is the uniquely sum of a scalar and a nilpotent (so nilpotents constitute the only maximal ideal of codimension 1). In other ...
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