# Questions tagged [ag.algebraic-geometry]

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

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### Non-noetherian schemes with noetherian underlying space (in the Zariski topology)

I am curious if there are many non-Noetherian schemes with Noetheiran underlying space. I know an example, we consider: $R= k[x_1, x_2, \dots]/\langle x_1^2, x_2^2, \dots\rangle$ for $k$ a field. ...
1 vote
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### Fitting closed polynomial curves to given points in the plane

Consider the equation $$x^2 + y^2 - R^2 = 0 \tag{1}\label{470162_1}$$ which describes a circle. For a given point (e.g. $P_1 = (0.8, 1.5)$) the parameter $R$ can be easily determined. I am ...
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1 vote
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### Origin of the name Trace resp Integral symbol for the trace map of Dualizing Sheaf

Let $X \subset \mathbb{P}^n_k$ be a normal projective subscheme over $k$ of dimension $n$. The dualizing sheaf is in context of Serre duality a pair $(\omega_X,t)$ (which exists in that case) ...
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• 21.7k
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### Normalizer of Levi subgroup

Let $G$ be a reductive group (we can work on an algebraically closed field if needed) and let $L$ be a parabolic subgroup, i.e. the centralizer of a certain torus $T \subseteq G$. Associated with this ...
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520 views

### Zeros of a function defined on $\mathbb{S}^2 \times \mathbb{S}^2$

Let $u$ be a smooth function on the sphere, and for each $y \in \mathbb{S}^2$, let $R_y$ be the $180^\circ$ rotation about the vector $y$. For each pair $(x, y) \in \mathbb{S}^2 \times \mathbb{S}^2$, ...
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### Reference request The support of $f$-nef divisor

I'm seaching for a proof of the theorem below. Do you know any reference? Consider $f:X\rightarrow Y$ projective birational map between normal varieties and $\mathbb{R}$ cartier divisor $D$ whose ...
• 199
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### Count N-tuples of commuting matrices over $F_q$ is given by polynomials with pattern $\sum q^{A_i(N)} P_{i}(q)$, where $P_i$ - do not depend on $N$?

Count pairs of $k \times k$ commuting matrices over finite field $F_q$ is given by certain polynomials in $q$ (which is quite rare phenomena for algebraic varieties) and have interesting generating ...
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### Finding rational points on intersection of quadrics in affine 3-space

Consider the subvariety of Spec $\mathbb{Q}[x,y,z]$ cut out by the equations \begin{eqnarray*} f_1: a_1x^2 - y^2 - b_1^2 & = & 0 \\ f_2 : a_2x^2 - z^2 - b_2^2 & = & 0 \end{eqnarray*} ...
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There is a theorem that (stable, topologically trivial) holomorphic $G$-bundles are in one-to-one correspondence to flat $K$-bundles (with the appropriate corresponding condition), where $K$ is the ...
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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 ...
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1 vote
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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 ...
227 views

### Sheaves which are locally free on subschemes 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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### 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 ...
1 vote
108 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 ...
237 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, ...
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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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### 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 $M_{1,1}$). ...
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1 vote
159 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 ...
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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 $g$ ...
120 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 ...
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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?
663 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,...
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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
1 vote
50 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 ...
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1 vote
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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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