# Questions tagged [ag.algebraic-geometry]

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

21,434
questions

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### how to generate the n-torsion group in an elliptic curve

Let $E$ be an elliptic curve over a field $K$.
I was curious about the following sentence: "then the $n$-torsion on $E(\overline{K})$ has known structure, as a Cartesian product of two cyclic ...

2
votes

0
answers

70
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### Why do we say IndCoh(X) is analogous to the set of distributions on X?

I've seen it written (for example, in Gaitsgory-Rozenblyum) that for a scheme $X$, the category IndCoh($X$) is to be thought of as a categorificaiton of the space of distributions on $X$, just as the (...

1
vote

1
answer

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### "General position" on $\mathbb{P}^1\times\mathbb{P}^1$

On $\mathbb{P}^2$ we have the notion of general positions: no 3 points on a line, no 6 on a conic, etc. In particular, blowing up points (up to 8) in general positions give ample anti-canonical class, ...

1
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0
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### Is there a name for a normal, projective variety where every effective divisor is ample?

Is there a name for a normal, projective variety such that every effective divisor is ample? Examples of such varieties are projective space, weighted projective spaces, and simple Abelian varieties ...

2
votes

1
answer

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### Polynomial roots

How can you decide, if a polynomial with integer coefficients p(x) is the product (q(x))^2*r(x) of two other polynomial with integer coefficients q(x), r(x)?

5
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1
answer

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### Operations on perverse sheaves on disk

The category of perverse sheaves on the disk is isomorphic to the category of diagrams
$$E\substack{\substack{c\\\to}\\\substack{v\\\leftarrow}}F$$
With $E,V$ finite dimensional vector spaces, and ...

4
votes

0
answers

107
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### Bezout-type theorem for $p$-adic analytic plane curves

Let $p$ be a prime, and let $f,g \in \mathbb{Z}_p[[x,y]]$ be power series convergent on all of $\mathbb{Z}_p$. Suppose that the intersection of the analytic plane curves cut out by $f$ and $g$ is ...

4
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0
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### $\pm 1$-equivariant perverse sheaves on the affine line

Let $G=\mathbb{Z}/\mathbb{2Z}$ act by the map $z\mapsto -z$ on a complex line $\mathbb{C}$. The category $\mathcal{Perv}(\mathbb{C})$ of perverse sheaves smooth along the stratification by the origin ...

1
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0
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### pullback of sheaves from reduced schemes

Let $X$ be a non reduced noetherian scheme. Is there a way to recognize or characterize the coherent sheaves $\mathcal E$ on $X$ such that there exist a reduced noetherian scheme $Y$, a morphism $f:X\...

0
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1
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263
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### Derivations and ideals

Let $R$ be a regular local ring and $I$ and ideal of $R$. If $D$ is a derivation of $R$, let
$$\lambda_D:I/I^2\to R/I$$
be the composition of the restriction of $D$ to $I$ and the quotient map $R\to R/...

1
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1
answer

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### Locus where morphism has positive-dimensional fibers

Let $f:\mathbb{A}^n\to\mathbb{A}^n$ be a dominant morphism of degree $d$. Then there exists a subvariety $Y\subseteq\mathbb{A}^n$ such that the fibre of $f$ over $y$ is a zero-dimensional subvariety ...

6
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0
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### Looking for counterexamples: Are maximal tori in the automorphism groups of smooth complex quasiprojective varieties conjugate?

Let $X$ be a smooth quasiprojective variety over $\mathbb{C}$. It has a group of (algebraic) automorphisms $
\DeclareMathOperator{\Aut}{Aut}
\Aut(X)$.
Define a torus in $\Aut(X)$ to be a faithful ...

21
votes

6
answers

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### A ring such that all projectives are stably free but not all projectives are free?

This question is motivated by this recent question. Suppose $R$ is commutative, Noetherian ring and $M$ a finitely generated $R$-module. Let $FD(M)$ and $PD(M)$ be the shortest length of free and ...

0
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1
answer

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### Integer quadratic representation subject to discriminant minimization algorithm

Let $f(x)=ax^2+bx+c$ and $f(x)=n$. Is there an algorithm to choose $a,b,c$ such that the discriminant is minimized? Where $a,b,c,n,x$ are all integers.
More concretely, is there an algorithm to find $...

2
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0
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### formal smoothness for henselian thickening

Assume that $A,I$ is an henselian pair over $R$ and $X$ is a smooth $R$ scheme. can we say that $X(A)\to X(A/I)$ is surjective? I know that this is true if $X$ is affine(or even quasi-projective) or ...

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### JEE Level Advanced Question [closed]

If m is the slope of a line which is tangent to y^3=x^4 and a normal to x^2-2x+y^2=0, then m is equal to:

1
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1
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### Nonequidimensional birational Mori contractions

I have been looking for an excplicit example of a birational, divisorial Mori contraction such that the exceptional locus is not equidimensional onto its image.
To agree with the setup I like, the ...

6
votes

1
answer

513
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### Different definition of Cox rings

Definition: Let $X$ be a normal projective variety with finitely generated Picard group. Define the Cox ring of $X$ as the multisection ring $$\text{Cox}(X)=\bigoplus_{(m_1,\ldots,m_k)\in \mathbb{N}^k}...

1
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1
answer

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### Characterisation of (integrable) connections on (trivial) principal bundle

Let $M$ be a manifold. Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra.
Let $P(M,G)$ be a principal bundle. Recall that, a connection on $P(M,G)$ is a distribution $\mathcal{H}\subseteq ...

2
votes

0
answers

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### Different definitions of the thick affine flag variety

I have seen several different definitions of the so called "thick" affine flag variety associated to an affine Lie algebra, and I am having trouble seeing why they are the same.
Some ...

5
votes

1
answer

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### Definition of discrepancy

In Kollar and Mori "Birational geometry of algebraic varieties" discrepancy is defined as following way.
Let X be a normal variety and $D = \sum_i a_i D_i$ be a $\mathbb{Q}$ divisor. Assume ...

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2
answers

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### What are the different theories that the motivic fundamental group attempts to unify?

I must preface by confessing complete ignorance in the subject. I've read introductory texts about the theory of motives, but I am certainly no expert.
In http://www.math.ias.edu/files/deligne/...

3
votes

1
answer

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### Čech cohomology refinement mapping

Let us consider the map $t_{AB}^*:H^1(A,F)\to H^1(B,F)$ between the cohomology groups, induced by the refinement map $t_{AB}:J\to I$, where $F$ is a sheaf of abelian groups on $X$, $A$ and $B$ are ...

2
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0
answers

692
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### Characterizing zeros of Schur functions over $\mathbb{R^n}$ or $\mathbb{C^n}$

Let $m_i\in \mathbb{N}, 1\leq i \leq n $ such that wlog if $m_i < m_j\in \mathbb{Z^+}$ then $i < j$.
Consider the Vandermonde matrix $$V=
\begin{bmatrix}
1 & 1 & 1 & \ldots & 1 ...

1
vote

0
answers

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### Hopf algebra from Chow rings of Hilbert schemes of smooth surface

Let $X$ be a smooth projective surface. As its Hilbert schemes of points are resolutions of the symmetric powers, the addition map $S^nX \times S^mX \rightarrow S^{n+m}X$ lifts rational map $X^{[n]} \...

5
votes

2
answers

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### Projective closure of affine curve

Is there a generalized method to find the projective closure of an affine curve? For example, I read that the projective closure of $y^2 = x^3−x+1$ in $\mathbb{P}^2$ is $y^2z = x^3−xz^2+z^3$.
If I ...

3
votes

1
answer

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### Action on étale fundamental group is conjugation

Let $X$ be a quasi-compact, quasi-separated connected scheme and let $\bar{x}$ be a geometric point. Denote by $\pi_1(X,\bar{x})$ the étale fundamental group, defined as the automorphism group of the ...

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0
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### Constructing curves with large tangent space in complex variety

Suppose $M$ is a (singular) complex analytic/algebraic variety. Then for every $p\in M$ there exists a (possibly reducible) curve $C \subset U\subseteq M$ containing $p$ such that $T_pC=T_pM$, where $...

1
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0
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### Regarding the common zeros of the system of equations

Consider the following two systems of n homogeneous polynomials in n variables of degree $d$ with complex coefficients:
System 1 ($S_1$):
$f_1(x_1,\dots,x_n) = 0$,
$f_2(x_1,\dots,x_n) = 0$,
$\vdots$
$...

0
votes

1
answer

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### On the solutions of system of homogeneous polynomials of degree $d$ in $n$ variables

Consider the following two system of n homogeneous polynomials in n variables of degree $d$ with complex coefficients:
System 1 ($S_1$):
$f_1(x_1,\dots,x_n) = 0$,
$f_2(x_1,\dots,x_n) = 0$,
$\vdots$
$...

6
votes

0
answers

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### Does there exist a plane curve such that it has the heart curve as catacaustic?

Given a curve $C$ and a fixed point $L$ (the light source), the catacaustic of $C$ with respect to $L$ is the envelope of light rays coming from $L$ and reflected from the curve $C$.
The catacaustic ...

4
votes

0
answers

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### Every stable homotopical functor factors through $\mathbf{SH}$

In this nlab page, it says that the fact that every stable homotopical functor factors through $\mathbf{SH}$ (the motivic stable homotopy category of Morel-Voevodsky) is proven in Ayoub's thesis. ...

3
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0
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### Splitting of normal bundle exact sequence and Holomorphic neighbourhood retract

Let $X$ be a compact complex manifold and $Y\subset X$ a complex submanifold of $X$.
Consider the two following conditions:
The exact sequence $0\to TY\to TX|_{Y}\to N_Y\to 0$, where $TX$, $TY$ ...

2
votes

2
answers

381
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### What is the pull-back of a polarization of abelian schemes over different bases?

The following came up when reading the definition of the moduli stack of principally polarized abelian varieties in [1].
Let $\pi_1:A_1 \to S_1$ and $\pi_2: A_2 \to S_2$ be abelian schemes over $S_i$, ...

2
votes

0
answers

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### Equality case of the log-Bogomolov-Miyaoka-Yau inequality

The Bogomolov-Miyaoka-Yau inequality for sufaces says that if $X$ is a smooth projective minimal surface of general type then $c_1(X)^2 \le 3 c_2(X)$. It is a theorem of Yau (I think) that equality ...

3
votes

1
answer

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### Approximating $p$-adic power series by polynomials

Let $p$ be a prime, and let $f \in \mathbb{Z}_p[[x_1,\dots,x_d]]$ be a power series convergent on all of $\mathbb{Z}_p^d$. We make the following definition concerning the approximation of $f$ by ...

6
votes

0
answers

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### Is every free additive action on the affine space conjugate to a translation?

Is every free action of the additive group $\mathbb{G}_a$ on the affine space $\mathbb{A}^3$ conjugate to a translation?
In characteristic zero, the answer is yes, and is due to Kaliman. [Kaliman, S. &...

5
votes

2
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### Are continuous rational functions arc-analytic?

Let $X\subseteq\mathbb{R}^n$ be a smooth semi-algebraic set (for simplicity we can assume $X=B(0,r)$ is a small ball around the origin). A function $f:X\rightarrow \mathbb{R}$ is called a continuous ...

0
votes

1
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### common zeroes of multivariable polynomials

Let $P_1(X,Y),\cdots,P_n(X,Y)$ be polynomials of $\mathbb C[X,Y]$ not all zero and $S$ be an infinite subset of $\mathbb C^2$ such that $P_1,\cdots,P_n$ vanish on $S$. My question:
do there exist a ...

3
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0
answers

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### Relationship between $\infty$-categories of ind-coherent sheaves on the base and total space

My question is vaguely as follows, let $G\to E\to X$ be a principal $\infty$-bundle for some group object $G$ in a category of ($\infty$-)stacks. Can we recover the category $\operatorname{IndCoh}(E)$ ...

2
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0
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### Kodaira-Spencer morphism - complete deformations

Let $X$, $T$ be smooth varieties over $\mathbb C$, $X$ projective, and $\mathcal E$ a coherent sheaf on $X\times T$, flat on $T$. Let $t_0\in T$ be a closed point. Suppose that all the sheaves ${\...

0
votes

1
answer

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### relations between non-negativity of multivariate polynomials and SOS over gradient ideal

We know there is a necessary condition for the non-negativity of multivariate polynomials in the paper "Sum of Squares Decompositions of Polynomials over their Gradient Ideals with Rational ...

4
votes

1
answer

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### Universal property of the category of quasicoherent sheaves of a blowup

We know that if $Z \rightarrow X$ is a closed subscheme of X of ideal $\mathcal{I}$, then if $\pi : Bl_Z X \rightarrow X$ is the projection, $\pi^* \mathcal{I}$ is invertible. Does the category of ...

3
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0
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### Derived pushforward of a Schur functor, and bounded derived categories of Grassmannians

Consider Grassmanianns over fields of characteristic zero.
Let $i : Gr_{k-1,n} \rightarrow Gr_{k,n+1}$ be the `direct sum' map between Grassmannians. By universal property of Grassmannian, this map ...

1
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0
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### Amitsur's theorem for generalized Severi–Brauer varieties

Let $k$ be a field of characteristic zero and assume that $A$ is a central simple algebra of index $2^n > 2$. We denote by $\operatorname{SB}_i(A)$ the $i$-th (generalized) Severi–Brauer variety of ...

2
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1
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### Motivation of Zariski–Van Kampen theorem

The Zariski–Van Kampen theorem gives the presentation of the fundamental group of the complement of the plane curve of degree $d$. But what's the motivation of this theorem? More generally, why are ...

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2
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### Classification of singularities of plane curves of fixed degree (reference request)

We know the answers to some questions like What is the maximal number of singularities of (reduced) plane curves of degree $d$? for general $d$ (in this case $\tfrac{1}{2}d(d-1)$, obtained by $d$ ...

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0
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### Number of conditions imposed by general points

I encountered with a problem when I read the part of Enriques-Babbage Theorem of the book Geometry of Algebraic Curves Vol. I by ACGH. It is stated on page 112-113 that all subsets of $m$ points of a ...

31
votes

1
answer

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### What is the Hirzebruch-Riemann-Roch formula for the flag variety of a Lie algebra?

If we have a finite dimensional Lie algebra g, then the flag variety of g is a projective scheme.
My question is what is Hirzebruch-Riemann-Roch formula for this projective scheme? Are there any ...

7
votes

0
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### Is a smooth projective variety over $\mathbb{C}$ dominated by a Ball?

Suppose that $X$ is a smooth projective variety of dimension $d$ over the complex numbers.
Is it true that there is a ball $\Delta_d=\{ z\in \mathbb{C}^d / \lvert z\rvert<1\}$ and a surjective ...