Questions tagged [ag.algebraic-geometry]

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

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4 votes
3 answers
1k views

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 views

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
114 views

"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 vote
0 answers
69 views

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
311 views

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 votes
1 answer
286 views

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 views

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 votes
0 answers
124 views

$\pm 1$-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 vote
0 answers
79 views

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 votes
1 answer
263 views

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 vote
1 answer
105 views

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 votes
0 answers
94 views

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
3k views

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 votes
1 answer
114 views

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 votes
0 answers
67 views

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 ...
-5 votes
0 answers
40 views

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 vote
1 answer
91 views

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 views

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 vote
1 answer
769 views

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
59 views

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
761 views

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 ...
19 votes
2 answers
3k views

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
157 views

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 votes
0 answers
692 views

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
78 views

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
8k views

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
568 views

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 ...
1 vote
0 answers
176 views

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 vote
0 answers
82 views

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
122 views

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
140 views

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
143 views

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 votes
0 answers
110 views

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 views

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
90 views

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
150 views

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
249 views

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 answers
354 views

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 answer
118 views

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 votes
0 answers
122 views

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 votes
0 answers
123 views

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
29 views

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
130 views

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 votes
0 answers
97 views

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 vote
0 answers
306 views

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 votes
1 answer
227 views

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 ...
11 votes
2 answers
1k views

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 $d$ ...
1 vote
0 answers
92 views

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
3k views

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 answers
222 views

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 ...

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