Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [ag.algebraic-geometry]

for questions on algebraic geometry, including algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

2
votes
0answers
115 views

What is an example of a cokernel $B/\phi(A)$ in group schemes which does not have $A=\mu_d$ and requires the fppf topology to be a sheaf?

Let $S$ be affine. A bit of background: Let us think of $S$-group schemes as abelian sheaves over a given site (etale, Zariski, fppf, etc). When we take a cokernel of a morphism $\phi$ this category: $...
3
votes
1answer
124 views

Can we move curves which are members of very ample systems?

Let us take the second degree Hirzebruch surface F_2 which is a holomorphic CP^1 bundle over CP^1 having sections of self intersections +2 and -2. Let me denote the class of the -2 section by C and ...
3
votes
0answers
113 views

Mixed Hodge Polynomial for Algebraic Stacks

Let $X$ be a complex algebraic variety. The numerical invariants associated with the Mixed Hodge Structure of $X$ can be encoded in a polynomial in three variables called the mixed Hodge polynomial $H(...
3
votes
0answers
106 views

Moduli space of nilpotent Lie algebras

Fix a nilpotent Lie algebra $L$ over some char 0 field $k$ which is naturally graded, i. e. isomorphic to graded algebra $\bar L$ associated to lower central filtration. I'm interested in some ...
1
vote
0answers
86 views

Separable extensions & topology vs inseparable extensions and algebra

In the note Properties of fibers and applications, Osserman writes above Definition 1.5: Intuitively, the point is that phenomena relating to topology can only change under separable extensions, ...
1
vote
0answers
77 views

Example of a spherical homogeneous space $G/H$ with a pairs of colors and with the center of $G$ not contained in $H$?

Let $G$ be a simply connected simple algebraic group over $\mathbb C$, $B\subset G$ a Borel subgroup, and $T\subset B$ a maximal torus. Let $\mathcal{S}=\mathcal{S}(G,T,B)$ denote the set of simple ...
5
votes
1answer
227 views

Confusion on summand of Hochschild homology of D-modules

I've encountered the following confusion. Start with the category of perfect D-modules on $\mathbb{A} ^{1}$, which I'll denote $D$. We have the object $\mathcal{O}$, which is exceptional in the sense ...
1
vote
1answer
119 views

Construction of log canonical singularity

I know there's classification about normal log canonical surface singularity in the sense of configuration of exceptional curves. There is one type of log canonical singularity(not klt) whose ...
4
votes
1answer
445 views

What are “arithmetic curves”?

Let $C$ be a separated irreducible reduced curve which is quasi-finite over $\mathrm{Spec}\: \mathbb Z$. Is it necessarily affine i.e. $\mathrm{Spec}\: \mathcal O$ where $\mathcal O$ is an order in a ...
1
vote
0answers
47 views

Schemes for conditional distributions (monads)

(Note: This is a heuristic question. I'm trying to work out if this idea makes sense. I don't have much background in this area, so apologies if I'm wide of the mark.) Suppose you have a monad ...
9
votes
0answers
130 views

Mixed characteristic analogue of algebraically of the diagonal of two-variable power series?

Let $f=\sum_{n,m \geq 0}^{\infty}[a_{nm}]p^ny^m \in \mathbb Z_p[[y]]$, where $a_{nm} \in \mathbb F_p$ and $[\cdot]$ means the Teichmüller lifting. Define $I(f)=\sum_{n \geq 0}[a_{nn}]p^nt^n \in \...
7
votes
0answers
133 views

Higher genus Gromov-Witten invariants and mirror symmetry

As a physicist, my understanding of mirror symmetry is very limited, and perhaps the most "mathematical" literature I have read on mirror symmetry is the book of M. Gross. In the genus-0 Gromov-Witten ...
2
votes
1answer
152 views

Vector bundles on henselian schemes

Let $X$ be a smooth and projective scheme over $\mathbf{Z}_p$. We call $\mathfrak{X}$ the ringed space whose topological space is the topological space of the special fiber of $X$, and whose ...
-2
votes
0answers
92 views

Is there always a curve inside $X$ with larger Neron-Severi group?

Let $X$ be a projective integral scheme over an algebraically closed field $k$. Does there exist a smooth projective curve $C\subset X$ such that the natural homomorphism $NS(X)\to NS(C)$ is injective?...
6
votes
2answers
319 views

Show Fiber Product of Rational Elliptic Surfaces is Calabi-Yau

In a handful of contexts people study Calabi-Yau threefolds formed by taking the fiber product of two rational elliptic surfaces. I can't find any detailed explanation of why such geometries are ...
1
vote
0answers
77 views

When a semigroup ideal is a determinantal ideal?

Let $S=\langle n_1,...,n_r \rangle$ be a commutative semigroup, and let $I_S \subset k[x_1,...,x_r]$ the associated ideal of $S$, defined as the kernel of the polinomial map $\varphi:k[x_1,...x_n] \...
2
votes
1answer
131 views

Purely inseparable isogeny

How to prove purely inseparable isogeny between two abelian varieties is radical (universally injective)? Purely inseparable morphism means the extension between the two function fields is purely ...
4
votes
1answer
182 views

Does there exist a curve which avoids a given countable union of small subsets?

Let $X$ be a projective variety over $\mathbb{C}$. Let $X_1, X_2, \ldots$ be proper closed subsets of $X$. Then $\cup_i X_i \neq X(\mathbb{C})$. However, I am interested in a stronger statement. ...
6
votes
1answer
259 views

Objects with trivial automorphism group

Suppose I am working with a category of objects such that each object $X$ in the category has as a trivial automorphism group. An example of such an object would be genus zero curves with $n$-...
5
votes
1answer
135 views

tangent cone and local picture

Let $X\subset \mathbb P^n_{\mathbb C}$ be a closed algebraic subset and $x\in X$. Suppose tha tangent cone of $X$ at $x$ is the union of (say) a plane and a line (meeting at the origin). Can we ...
3
votes
0answers
87 views

Primary decomposition for noetherian schemes

I'm trying to understand the natural scheme theoretic structure for the irreducible components of a noetherian scheme $X$. The idea would be that if $\text{Ass}(\mathcal{O}_X)=\{x_1,\dots x_n\}$ ...
2
votes
0answers
163 views

Derived category and flatness

It is well known that if two varieties are Fourier-Mukai partners, then there are strong constraints on them. For example, if one of them is Calabi-Yau the other one must be Calabi-Yau too. Similarly ...
2
votes
1answer
84 views

Core components of quiver varieties as fiber bundles of flag varieties

Is there an example of Nakajima quiver variety of type A which has all core components smooth, such that at least one of them is NOT an iterated fibre bundle of flag manifolds (i.e. a space obtained ...
5
votes
1answer
105 views

Failure of Borel-Schmid quasi-unipotency theorem in non-algebraic case

Let $\mathcal{X} \to \Delta^{\times}$ be a smooth family of compact Kähler manifolds over a punctured disc. When this family is algebraic, the celebrated quasi-unipotency theorem (I think, due to ...
0
votes
0answers
64 views

Properties of a particular Kummer Surface

Let $Y$ be the abelian variety $\mathbb{C}/\mathbb{Z}[i] \times \mathbb{C}/\mathbb{Z}[i] $ where $\mathbb{Z}[i]$ denote the set of Gaussian integers in $\mathbb{C}$. Let $X$ be the quotient of $Y$ by ...
0
votes
0answers
59 views

Definition of weak normality

Let $X$ be a variety over a field $F$ of characteristic $p>0$. The usual (not 100% sure) definition of weak normality should be $X$ is weakly normal if whenever $f:Y\to X$ is a morphism of ...
2
votes
1answer
139 views

Some naive questions on crepant resolutions of singularities

I have some clarifying questions about crepant resolutions of singularities. The definition that I am aware of is that if $f:\tilde X \to X$ is a resolution of singularities, then the resolution is ...
8
votes
0answers
158 views

Hyperkähler structure on the moduli space of tetrahedra?

Consider a moduli space of geodesic tetrahedra in the hyperbolic space $\mathbb{H}^3.$ In the Klein's model the hyperbolic space can be presented as the interior of a unit ball $$ \mathbb{H}^3=\{(x_1,...
3
votes
0answers
65 views

Degeneration of a fiber of a rational elliptic surface

Let $\mathcal{E}$ be a rational elliptic surface, obtained by blowing up nine base points of a pencil of elliptic curves in $\mathbb{P}^2.$ Possible singular fibers of the elliptic fibration $|-K_\...
2
votes
0answers
107 views

field of definition of CM abelian varieties

When $A$ is a CM abelian variety of dimension $1$ (i.e., an elliptic curve), then we have a result that if it has CM by a maximal order then it has a model over a number field $F$ where $F$ is the ...
1
vote
0answers
97 views

Lift up characteristic class to chain complex

In derived category, there is a slogan, "cohomology is bad, chain complex is good". In the theory of characteristic classes, we could associate a vector bundle to cohomology classes of the base space. ...
0
votes
0answers
85 views

How is this pairing $\langle\,,\rangle$ defined of cocharacter and character of an algebraic group?

Let $G$ be a semisimple linear algebraic group. Let $X^*$ be the group of characters and $X_*$ be the group of cocharacters. Then I know that there exists a pairing $\langle\,,\rangle : X^*(G) \times ...
1
vote
0answers
50 views

Strongly isovariant (aka fixed point reflecting / stabiliser preserving) morphisms

I have some questions about what feels like basic topics in quotients of schemes by group actions. Consequently I suspect there are well-known references; I couldn't find them, though. First ...
13
votes
1answer
708 views

Applications of derived categories to “Traditional Algebraic Geometry”

I would like to know how derived categories (in particular, derived categories of coherent sheaves) can give results about "Traditional Algebraic Geometry". I am mostly interested in classical ...
2
votes
0answers
90 views

Intersection of a reduced projective variety with a general hyperplane is reduced

Let $X\subset \mathbb{P}^n$ be a reduced closed subscheme. For a general hyperplane $H$, $X\cap H$ is again reduced (and of dimension one less). Is there an easy proof of this result? Algebraically, ...
7
votes
0answers
251 views

Deligne conjecture without Langlands correspondence

Let $X$ be a normal variety over a finite field $F_q$. Fix a prime number $l$ relatively prime to $q$. Let $\sigma$ be a irreducible lisse $l$-adic sheaf on $X$ whose determinant has finite order. It ...
3
votes
1answer
328 views

Coordinate free expression for the determinant of a $2 \times 2$ matrix

Let $E\rightarrow X$ be a rank two vector bundle on a variety $X$. How can one write an abstract map $Sym^2(E)\rightarrow (\wedge^2 E)^{\otimes 2}$ that in local coordinates, when we fix a frame of $E$...
2
votes
0answers
126 views

Extension of alg closed fields = limit of smooth algebras

In the proof of https://stacks.math.columbia.edu/tag/0F0B, it is a claim that if $K/k$ is an extension of algebraically closed fields, then $K$ is a limit of smooth $k$-algebras. This is justified ...
-1
votes
0answers
73 views

Proper projections

Let $D \subset \mathbb{C}^k$ be your favorite complex domain. Suppose we are given a proper holomorphic mapping $f \colon D \to \mathbb{C}^{k+2}$. Let us take $k+1$ generic linear functions $l_i \...
2
votes
0answers
142 views

Splittings in the difference bundle construction of Atiyah-Hirzebruch

I'm reading the construction of difference bundle(generalized) in the paper of Atiyah-Hirzebruch(Analytic cycles on complex manifolds) There is one thing I cannot understand. The followings are in ...
1
vote
0answers
59 views

Orbits of unipotent groups

Let $V$ be a real vector space (of finite dimension) and let $G$ be a unipotent Lie subgroup of $\mathrm{GL}(V)$. The orbits of points under the action of $G$ (that is, the sets $Gx = \{T(x) \ : \ T \...
9
votes
0answers
132 views

Hilbert schemes of points on toric surfaces

Let $\mathrm{S}$ be a smooth toric surface. The Hilbert scheme of $n$ points $\mathrm{Hilb}^n(\mathrm{S})$ inherits a torus action, but need not admit the structure of a toric variety itself. For ...
3
votes
1answer
93 views

Subbundle generated by linearly dependent sections

On $\mathbb{P}^1$ consider the trivial bundle $\mathcal{O}\oplus \mathcal{O}$, and the subbundle $\mathcal{L}_{a,b}\subset\mathcal{O}\oplus \mathcal{O}$ that on an open subset $U$ of $\mathbb{P}^1$ is ...
9
votes
1answer
273 views

How is the sheaf defined for $G/H$ where $G$ is an algebraic group and $H$ is a normal closed subgroup?

I have learned that if $G$ is an algebraic group and $H$ is a normal closed subgroup then $G/H$ is also an algebraic group satisfying: for any morphisms $\phi : G \rightarrow X$ constant on the ...
4
votes
0answers
128 views

lemma II.2.4 in Harris-Taylor (about drinfeld-katz-mazur level structure on 1-dimensional $p$-divisible groups)

Lemma II.2.4 on page 82 in Harris and Taylor's "The Geometry and Cohomology of Some Simple Shimura Varieties" (or lemma 3.2 here), says that given a Drinfeld(-Katz-Mazur) level structure $\alpha:(p^{-...
-5
votes
0answers
224 views

How to show $g$ is projective?

The above pages are from Algebraic Geometry by Lei Fu. Here a morphism $\varphi: Z\to Y$ is projective if $\varphi$ can be factorized as $Z \to \mathrm{P}_Y^n\to Y$ for some closed immersion $Z\to \...
1
vote
1answer
149 views

Very ample linear systems - intersections with multiplicity >1

On a degree $n$ Hirzebruch surface $F_n$, suppose we have a very ample linear system. It is known that its generic smooth irreducible members give a Lefschetz pencil on $F_n$. Let us take a member, $G$...
21
votes
1answer
444 views

What automorphic forms are expected to occur in the zeta function of moduli space of curves?

Assume $g \geq 1$ and $n \geq 0$, the moduli stack ${\mathcal {M}}_{g,n}$ classifies families of smooth projective curves of genus $g$ with $n$ marked points , together with their isomorphisms. It has ...
9
votes
0answers
263 views

Gauge theory on schemes

Gauge theories are traditionally formulated in space-time, superspace, manifolds, or supermanifolds. Are there formulations of gauge theory in the context of algebraic geometry (e.g. defining fields ...
0
votes
0answers
38 views

Recovering integrality by solving rational polynomial systems that approximate real polynomial systems

Given $n$ polynomials $g_1(x_1,\dots,x_{n}),\dots,g_{n}(x_1,\dots,x_{n})\in\mathbb R[x_1,\dots,x_{n}]$ where each of $g_1(x_1,\dots,x_{n}),\dots,g_{n}(x_1,\dots,x_{n})$ is homogeneous of degree $d$ ...