# Questions tagged [ag.algebraic-geometry]

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

21,450 questions
Filter by
Sorted by
Tagged with
54 views

### 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 $\mathscr{X}$ ...
• 697
53 views

### 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 ...
• 19
80 views

### 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?
• 19
117 views

### 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 ...
135 views

### 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 ...
• 1,051
109 views

### 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 ...
• 5,786
65 views

### 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 ...
• 143
107 views

### 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 ...
143 views

### 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 ...
• 333
1 vote
62 views

### 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 ...
128 views

• 141
79 views

### 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 $R^1f_*\mathcal{O}_X(mK_X)$ ...
62 views

• 721
100 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 ...
• 1,171
114 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 ...
• 493
82 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 ...
• 65
1 vote
98 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 ...
60 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 ...
• 383
1 vote
82 views

1 vote
83 views

• 721
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 ...
• 59
98 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 ...
• 183
1 vote
93 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 ...
• 383
131 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 ...
223 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 ...
• 163
249 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 ...
• 181
127 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 $X=\Tot(L)$...
• 139
207 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 ...
• 1,342
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 \$...