# Questions tagged [ag.algebraic-geometry]

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

21,450
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### Description of pull-back of coherent sheaves under a smooth morphism of Artin stacks

I am new to these formalisms, so pardon me if the question is basic. Let $\mathscr{X}$ be an Artin stack (you can take it to be Deligne-Mumford stack if it helps). By a coherent sheaf on $\mathscr{X}$ ...

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### Braches and the Radius of Convergence [closed]

I have the impression that Newton's Polygon makes precise the intuitive notion of the branches of an algebraic finction and gives a convergent series corresponding to each branch. Is my impression ...

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### Newton's Parallelogram

George Chrystal discusses Newton's Parallelogram. (Page 386, Part 2, Algebra: An Elementary Text-book.) Why is Newton's Polygon also called Newton's Parallelogram?

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### Height of a conductor ideal

We say an extension of Noetherian rings $R\subset S$ is elementary subintegral if $S=R[b]$ for some $b\in S$ with $b^2,b^3\in R$. The conductor ideal is defined to be $\operatorname{Ann}_R(S/R)$. What ...

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### Reference request: Automorphisms of $\mathbb C\{x,y\}$ which preserve the equation of the cusp, $x^3 - y^2$

In my research I encountered automorphisms of the ring of convergent power series
$$\varphi: \mathbb C\{x,y\} \to \mathbb C\{x,y\},$$
which preserve $f = x^3 - y^2$, i.e. $\varphi(f) = f$. I'm ...

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### Question about pro-representable Automorphism functor in Sernesi's "Deformations of algebraic schemes" with trivial relative Tangent space

Let $k$ be an algebraically closed field of arbitrary characteristic, $R$ a complete local noetherian, but non Artinian (see #Edit below for justification) $k$-algebra with residue field $k$ and $S$ a ...

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### Classification of principally polarized abelian surfaces - reference request

I found in Encyclopedia of math
https://encyclopediaofmath.org/wiki/Abelian_surface
there is a claim that:
"A principally polarized Abelian surface $(A,λ)$ is either the Jacobi variety $J(H)$ of ...

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107
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### Noether's formula for real algebraic surfaces

Is there a version of Noether's formula for the Euler characteristic of a surface for Real algebraic surfaces?
Specifically, given $X$ a real algebraic compact smooth surface, what is the relationship ...

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### What should be unipotent de Rham homotopy group?

What exactly should unipotent $\pi_1^\text{dR}$ be conceptually? What formal properties should it satisfy? This seems to be answered by Chen's theorem, which is stated in Corollary 3.269 of Multiple ...

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### Unique polarization on a very general curve with Mumford-Tate

I try to understand why a very general curve (smooth, projective over $\mathbb{C}$) has an unique polarization up to scalar on the $H^1(X,\mathbb{Q})$.
I was advised to look at the maximality of the ...

4
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### Is the asymptotic rank of a tensor bounded by (naive) border rank?

Write $R(T)$ for the rank of an order-$3$ tensor $T \in \mathbb C^{m \times n \times p}$ over the complex numbers. If $T' \in \mathbb {C}^{m' \times n' \times p'}$ is another such tensor then let $T \...

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### Degeneracy maps of Drinfeld modular curves

Over number fields, we have two natural degeneracy maps
$$X_0(N)\leftarrow X_0(pN) \rightarrow X_0(N)$$
between the (compactified) moduli space of elliptic curves with level $pN$ and $N$ respectively (...

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72
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### About the finite generation of log canonical rings in BCHM

I have posted this question on MSE but haven't received an answer yet. I rephrase it here.
Let $(X,B)$ be a klt pair where $K_X+B$ is $\mathbb{R}$-Cartier. Let $\pi:X\rightarrow U$ be a projective ...

1
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1
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### About the semi-ampleness for $\mathbb{R}$-divisor

Let $X$ be a normal proper variety and $D$ an $\mathbb{R}$-Cartier divisor on $X$. Then $D$ is called a semi-ample $\mathbb{R}$-divisor if there is a morphism $f:X\rightarrow Y$ and a ample $\mathbb{R}...

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### Are the higher direct images of pluricanonical bundles torsion-free?

Suppose $f: X\rightarrow S$ is a projective smooth morphism to a smooth variety $S$. Let $m\geq 2$ be a natural number.
It is known that the first higher direct image sheaf $R^1f_*\mathcal{O}_X(mK_X)$ ...

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### Potential typo in "Complete Systems of Two Addition Laws for Elliptic Curves" by Bosma and Lenstra

Here is a link to the article: https://www.sciencedirect.com/science/article/pii/S0022314X85710888?ref=cra_js_challenge&fr=RR-1.
Pages 237-238 give polynomial expressions $X_3^{(2)}, Y_3^{(2)}, ...

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### Interpretation of model theory in algebraic geometry

I found a paper Some applications of a model theoretic fact to (semi-) algebraic geometry by Lou van den Dries. In this paper, the author uses model theoretical methods to prove the completeness of ...

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### Nice proof that de Rham complex computes Lie algebra cohomology?

If $G$ is a nice enough group acting on a nice enough space $X$, then the relative de Rham complex
$$\Omega^\bullet_{X/(X/G)}\ \simeq\ \mathcal{O}_X\otimes\text{Sym}\,\mathfrak{g}^*[-1]$$
is given by (...

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

$\DeclareMathOperator\IndCoh{IndCoh}\DeclareMathOperator\QCoh{QCoh}$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 ...

3
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1
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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, ...

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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 ...

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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 ...

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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\...

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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 ...

2
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1
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114
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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 ...

2
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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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### 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 ...

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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 ...

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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]} \...

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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 ...

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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 $...

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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 ($S_1$):
$f_1(x_1,\dots,x_n) = 0$,
$f_2(x_1,\dots,x_n) = 0$,
$\vdots$
$...

4
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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. ...

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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 $TX$, $TY$ ...

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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
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1
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151
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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 ...

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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 ...

0
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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 ...

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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. &...

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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 $\operatorname{IndCoh}(E)$ ...

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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
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1
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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 ...

3
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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 ...

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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 ...

4
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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 ...

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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 ...

4
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### 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
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127
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### 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 $X=\Tot(L)$...

3
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### 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 ...

12
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1
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### 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 $...