Questions tagged [ag.algebraic-geometry]
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
21,607
questions
3
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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 ...
1
vote
0
answers
71
views
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 ...
2
votes
1
answer
108
views
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, ...
2
votes
1
answer
158
views
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}=...
3
votes
1
answer
206
views
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}$).
...
1
vote
1
answer
123
views
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 ...
2
votes
1
answer
131
views
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$ ...
2
votes
0
answers
67
views
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 ...
5
votes
1
answer
141
views
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) @&...
0
votes
0
answers
46
views
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?
-1
votes
0
answers
43
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What are the relations among canonical basis and dual canonical basis of the group of $A_4$?
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 ...
5
votes
2
answers
575
views
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,...
-6
votes
0
answers
47
views
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 ...
0
votes
1
answer
179
views
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 ...
-1
votes
0
answers
54
views
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 ...
1
vote
0
answers
36
views
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 ...
1
vote
0
answers
65
views
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 ...
1
vote
0
answers
66
views
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 ...
2
votes
0
answers
188
views
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 ...
0
votes
0
answers
56
views
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 ...
0
votes
0
answers
93
views
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 ...
2
votes
2
answers
422
views
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 ...
0
votes
1
answer
214
views
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 ...
2
votes
0
answers
209
views
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 ...
4
votes
0
answers
250
views
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, ...
2
votes
0
answers
78
views
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? ...
3
votes
0
answers
75
views
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-...
5
votes
0
answers
193
views
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 ...
2
votes
0
answers
94
views
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
vote
0
answers
50
views
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
votes
0
answers
147
views
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 ...
2
votes
1
answer
132
views
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 ...
0
votes
1
answer
104
views
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 ...
4
votes
0
answers
160
views
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\...
2
votes
0
answers
124
views
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 ...
4
votes
1
answer
219
views
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 ...
1
vote
0
answers
104
views
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 ...
2
votes
1
answer
151
views
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 ...
3
votes
0
answers
176
views
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 ...
1
vote
0
answers
114
views
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}...
4
votes
1
answer
233
views
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 ...
12
votes
1
answer
497
views
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 $\mathbb{Z}_p[[x]]$?
6
votes
1
answer
855
views
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 ...
3
votes
0
answers
193
views
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
vote
0
answers
84
views
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}...
1
vote
0
answers
157
views
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 ...
3
votes
1
answer
103
views
A question related to the strong Oda conjecture
A fan is a collection of strongly convex rational polyhedral cones in $\mathbb Z^n$, which we often think of as contained in $\mathbb Q^n$ or $\mathbb R^n$ for purposes of visualizing it. The defining ...
3
votes
1
answer
261
views
Is the subscheme parametrizing the k-th degeneracy loci Cohen-Macaulay?
Let $X$ be a Cohen-Macaulay projective variety (say over an algebraically closed field of characteristic 0), and $E,F$ two vector bundles such that $E^*\otimes F$ is globally generated. Consider the ...
0
votes
1
answer
126
views
Double disk bundle
Double disk bundle: A smooth, closed manifold
$M \cong DB^{-} \cup_L DB^{+}$
where
· $B^{±}, L$ smooth, closed manifolds
· $D^{l± +1} → DB^{±} → B^{±}$ smooth disk bundles such that
$S^{l±} → L \cong ...
2
votes
0
answers
62
views
Existence of meromorphic differential form on curve with given multiplicity of zeroes and poles
Let $m \in \mathbb{Z}^n$ be a partition of $2g-2$. Polishuk showed in his paper "Moduli spaces of curves with effective r-spin structures" (arXiv link) that if all entries of $m$ are ...