# Questions tagged [ag.algebraic-geometry]

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

14,707 questions

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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**

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**1**answer

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**

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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(...

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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 ...

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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, ...

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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**

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**1**answer

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 ...

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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**

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**1**answer

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 ...

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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 ...

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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 \...

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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**

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**1**answer

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 ...

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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**

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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 ...

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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] \...

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**1**answer

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**

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**1**answer

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.
...

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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 $n$-...

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**1**answer

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 ...

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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**

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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 ...

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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**

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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 ...

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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 ...

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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**

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**1**answer

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 ...

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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,...

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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_\...

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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 ...

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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. ...

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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 ...

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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**

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**1**answer

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 ...

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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, ...

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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 ...

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**1**answer

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 $E$...

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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 ...

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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 \...

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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 ...

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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 \...

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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 ...

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**1**answer

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**

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**1**answer

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 ...

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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^{-...

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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 \...

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**1**answer

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, $G$...

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**1**answer

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 ...

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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 ...

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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 $d$ ...