Questions tagged [ag.algebraic-geometry]

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

Filter by
Sorted by
Tagged with
3 votes
1 answer
178 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
695 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
82 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
573 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
187 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
85 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
133 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
155 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
154 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
125 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
382 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
99 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
158 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
257 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
361 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 ...
1 vote
1 answer
128 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
128 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
127 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 ${\...
4 votes
1 answer
139 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 ...
1 vote
0 answers
318 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
237 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
100 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
226 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 ...
12 votes
1 answer
2k views

Who proved the motivic 6-functor formalism?

In the recent beautiful talk "Motives and ring stacks" Peter Scholze states the theorem saying that there exists an initial 6-functor formalism on $\mathit{Sch}_\mathbb{Z}$ such that when $...
4 votes
0 answers
274 views

What is known about the number of elements needed to generate a given ideal in $k[X_1,\dots,X_n]$?

In Algebraic Geometry by J.S. Milne, after he proves Hilbert's Basis Theorem, he makes the following aside: One may ask how many elements are needed to generate a given a given ideal $\mathfrak a$ in ...
3 votes
0 answers
215 views

A Brauer group of a double covering of a "well-understood" variety

Let $k$ be a field (it is possible to assume that $k = \mathbb{Q}$ or $= \overline{\mathbb{Q}}$) and $X, Y$ nice varieties over $k$. Let $f \colon Y \to X$ be a finite flat surjective morphism of ...
3 votes
1 answer
193 views

Tangent space of a GIT quotient of $GL_{N}$

Let $G:=\operatorname{GL}_{N}$ act on its Lie algebra $\mathfrak{g}:=\mathfrak{gl}_{N}$ by conjugation. Then it acts naturally on the associated ring $\mathcal{O}(\mathfrak{g})$ of (algebraic or ...
5 votes
0 answers
231 views

Hyperelliptic curve with prescribed rational points?

Given a set of rational points $S$, does there always exist a hyperelliptic curve $C$ such that $C(\mathbb{Q})=S$? Namely, which sets could arise as the set of rational points of a hyperelliptic curve?...
3 votes
0 answers
132 views

Simple Grothendieck-Riemann-Roch computation with relative Todd class

$\DeclareMathOperator\Tot{Tot}\DeclareMathOperator\ch{ch}\DeclareMathOperator\td{td}\DeclareMathOperator\ker{ker}\DeclareMathOperator\rk{rk}$I was wondering if the following is correct: Let $X=\Tot(L)$...
1 vote
1 answer
423 views

Vector bundles on $\mathbb{P}^1$

I am considering an alternative proof of Grothendieck's classification of vector bundles on $\mathbb{P}^1$. Given a vector bundle $E$ on $\mathbb{P}^1$ one can associate a graded module $\Gamma(E)$ ...
1 vote
0 answers
87 views

About the relationship between Cayley-Chow families and well-defined family of proper cycles

I'm studying Chow varieties introduced in Chapter I.3-4 of "Rational curves on algebraic varieties" [Kol96] by János Kollár and also very interested in the "open" Chow variety ...
6 votes
1 answer
472 views

How to see that Eisenstein series are eigenfunctions of the laplacian?

Let $\Gamma$ be a discrete subgroup of $PSL_2(\mathbb{R})$ of finite type. Let $c_1,\ldots,c_h\in\mathbb{R}\cup\{\infty\}$ be a set of representatives of the $\Gamma$-equivalence classes of cusps. For ...
4 votes
0 answers
216 views

GIT quotient of a reductive Lie algebra by the maximal torus

Let $G$ be a connected complex reductive group with Lie algebra $\mathfrak{g}$. One knows a lot about the GIT quotient $\mathfrak{g}/\!/G$: the invariant ring is a free polynomial algebra on $\mathrm{...
2 votes
0 answers
162 views

Splitting of de Rham cohomology for singular spaces

I am currently trying to wrap my head around the following splitting result by Bloom & Herrera (here is a link to the ResearchGate publication) for the de Rham cohomology of (in particular) a ...
2 votes
1 answer
209 views

Higher direct images along proper morphisms in the non-Noetherian setting

Let $f : X \to Y$ be a finitely presented proper morphism. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Do the functors $R^i f_* \mathcal{F}$ preserve any of the following properties: (1) ...
1 vote
0 answers
143 views

Blowing up a reduced ideal in a normal variety

If I blow up a reduced ideal sheaf in a normal variety, is the resulting variety normal?
0 votes
0 answers
117 views

Nullstellensatz + Zariski Density?

My algebraic geometry is a little rusty, so sorry if this is really easy. Here is the situation I have: I have $n+1$ polynomials $p_1(x_1, x_2, \dots x_n, y)$, $p_2(x_1, x_2, \dots x_n, y)$, $\dots$ , ...
2 votes
0 answers
116 views

Integral geometric meaning of diameter

Let $X\subset \mathbb CP^n, n>2$ be a complex smooth algebraic hypersurface. Any hyperplane section $H\cap X$ is connected and has diameter $Diam(H\cap X)$ in the inner metric induced from the ...
1 vote
0 answers
185 views

Vakil's Generalization of qcqs Lemma

(This was also simultaneously asked on math stack exchange: https://math.stackexchange.com/questions/4857715/vakils-generalization-of-qcqs-lemma) In the most recent notes of Vakil, this is problem 15....
2 votes
1 answer
349 views

Can an abelian surface be bielliptic

Is an abelian surface containing an elliptic curve a bielliptic surface? Suppose I have an abelian surface $A$ over the complex numbers that contains an elliptic curve $E$. Then $A \to A/E$ is an ...
8 votes
0 answers
334 views

Tate's thesis and Riemann-Roch - $\mathrm{GL}_n$ or twisted version?

I recently learned why the Tate's thesis, especially Poisson summation formula, over a function field $F = \mathbb{F}_q(X)$ of a smooth projective curve $X_{/ \mathbb{F}_q}$ implies Riemann-Roch ...
2 votes
0 answers
120 views

Hodge coniveaux of Calabi-Yau manifolds

Let $X$ be a strict compact Calabi-Yau manifold of dimension $n$. By this, I mean that $X$ is a simply connected projective manifold whose holomorphic forms are generated by a nowhere zero top degree ...
3 votes
1 answer
184 views

Extending abelian schemes and their polarizations from an open subset

Let $S$ be a smooth quasi-projective scheme over $\mathbb{Z}$, and $A$ an abelian scheme over an open subset $U \subset S$. Suppose that $S\backslash U$ has codimension at least $2$ and that for every ...
11 votes
1 answer
1k views

Hodge conjecture as the equality of arithmetic and algebraic weights of motivic L-functions

Recently I became aware of the following statement given on page 13 of this paper. First, let us recall the following definitions: Definition 4.1. Suppose $L(s)$ is an analytic $L$-function with ...
1 vote
1 answer
232 views

Is this toric variety always smooth?

Let $k$ be an algebraically closed field. Let $\sigma$ be a 3-dimensional simplicial cone in $\mathbb{R}^3$ and let $\rho$ be a ray in $\sigma$. Let $U_{\rho}$ be defined as $\operatorname{Spec}(k[\...
1 vote
1 answer
227 views

Zeroes of elementary polynomials without involving closed-form solutions

Consider the following two polynomials, where $n$ is an integer: $$ p_n(x) = x^3-\frac1nx-\frac2n, \\ q_n(x) = x^2-\frac2n. $$ For any $n$, let $x_p=x_p(n)$ and $x_q=x_q(n)$ be the unique positive ...
0 votes
1 answer
242 views

Triple covers of $\mathbb{P}^2$ with Tschirnhausen module $\mathcal{O}(-1)\oplus\mathcal{O}(-1)$

Let $X$ be a surface as in the title. Rick Miranda said that $X$ is a Steiner cubic in $\mathbb{P}^4$, and the cover map is projection. Invariants of $X$ can be computed directly, $p_g(X)=0,K^2_X=8,e(...

1
3 4
5
6 7
431