Questions tagged [ag.algebraic-geometry]
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
21,603
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Subadditivity of Kodaira dimension
Given an algebraic fiber space $X \to B$ where $X$ and $B$ are smooth projective varieties over $\mathbb{C}$, it is known that the Kodaira dimensions satisfy the following subadditivity property:
$$\...
- 233
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114
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normal space of Brill--Noether variety
Let $C$ be a smooth projective curve, $J$ its Jacobian (of degree $d$, parametrizing degree $d$ line bundles, $d \geq 0$). Let $W_d^r$ be the Brill--Noether variety parameterizing degree $d$ line ...
- 131
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265
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Existence of a point on the Shimura variety of PEL-type correponding to a specific abelian variety
I have been puzzle by the following question for a while. Suppose that we have an a Shimura variety $Sh(G,h_0)$ given by some datatum $(L, V, \psi, h_0)$ such as in Section 4.9 of "Travaux de Shimura"...
- 570
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490
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differential forms on formal schemes
iam learning formal schemes. Suppose $Y\subset X$ are schemes, and $\hat{X}$ is the completion of $X$ along closed subscheme $Y$.
I wondered if there is a notion of sheaf of differential forms on a ...
- 21
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122
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Local positivity of an ample divisor
Let $X$ be a projective normal variety, $D$ be a Cartier divisor on $X$ and $A$ be an ample divisor on $X$. Let $x \in X$ be a (not necessarily closed) point. If the asymptotic vanishing order of $D$ (...
- 363
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355
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Nef and effective classes on abelian varieties
Is there any characterization of rational nef classes that don't come from effective $\mathbb{Q}$-divisors on abelian varieties? Is there any result along the lines of "Any nef $\mathbb{Q}$-divisor is ...
- 653
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227
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Is always a Crepant birational map between smooth varieties a small modification
Lemma: A crepant birational map between quasi-projective normal varieties with only terminal singularities is in fact small modifitacions, i.e. isomorphism in codimension 1.
So, if $f:X\...
- 325
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357
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one "big" Hilbert scheme?
I was recently reading one of the papers of Kapranov on $\mathcal{M}_{0,n}$ and he says that one can see it as a subvariety of the Hilbert scheme parametrizing all subschemes of a certain projective ...
- 3,717
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173
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Smoothing of a hyperquotient singularity
Let $f$ be a polynomial in $k$ complex variables, and suppose the affine variety $V$ given by $f = 0$ has an isolated singularity at the origin, but is otherwise smooth. Now assume that some cyclic ...
- 884
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56
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Combinatorical surface is restricted to a closed face an injection
Hello :)
I'm third year student of mathematics. In my own intrest i'm studying topology in combinatorical sense. Herefore i found also an lecture note in knot theory from Roberts. I want to understand ...
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141
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Relationship between stabilizers of a general point and a boundary point
Let $V$ be an n-dimensional complex vector space, and $u\in S^nV$ be a polynomial, $G(u)$ be the stabilizer of $u$ in $GL(V)$. Let $[v]\in\overline{GL(V)\cdot[u]}\subset\mathbb{P}(S^nV)$, but $v\notin ...
- 105
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328
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Darboux Surface
Let $D'$ be the set of degree $4$ smooth surfaces in $\mathbb{P}^3$, where for each $S\in D'$ there are 6 hyperplanes $H_1, \dots, H_6\subset\mathbb{P}^3$ in general positions, such that $S$ passes ...
- 105
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405
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a question about Beauville-Laszlo
Hi,
let $V$ be a complete DVR with uniformizer $\pi$. Let $m$ be a NON zero integer, $a\in V[[u,v]]/(uv-\pi)^{\times}$ and $f=\pi^{m}a$. Consider $F$ as the kernel of the diagram
$$
V[[u,v]]/(uv-\pi)...
- 11
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70
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reduced group covers of a curve
Let $C$ be a projective smooth connected curve over an algebraically closed field $k$. Let $(P,G,p)$ be a triple, where $G$ is a finite $k$-group scheme, $P$ is a $G$-torsor over $C$, $p\in P(k)$ a ...
- 304
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89
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field of coefficients, automorphism, cohomology
Let $k \subset \mathbb{C}$ be a number field and $X$ a smooth algebraic variety over $k$. Asssume $X$ is equipped with an automorphism $h$ of order $n$. Then $h$ induces linear maps in singular ...
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229
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Properties of quotient variety
Ok, so I have a projective variety $X$ over an algebraically closed field, and a finite group $G$ acting on $X$. Let $\pi:X\to Y=X/G$ be the quotient. Let $C$ be a curve in $Y$, and let $F=\pi^{-1}(C)$...
- 653
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272
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sections of vector bundles
Let $X$ a smooth projective connected curve over $\mathbb{C}$.
Let $E$ a vector bundle and $E'$ a subbundle of $E$.
Let $(x_{1},\dots,x_{n})$ $n$ closed points on the curve $X$ with $n>2g$ and $z$ ...
- 3,432
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274
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Topological invariance of intersection number
Let $X$ and $Y$ be algebraic curves in ${\mathbb P}^2$ and suppose they have an isolated intersection at $(0,0)$. Let $\hat{X}$ and $\hat{Y}$ be another such pair, and suppose that there exist ...
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293
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Conjugation of Dolbeault cohomology and cup product.
Let $H^p(X,\Omega_X^q)$ denote the (p,q) Dolbeault cohomology group of a Kähler manifold X. Conjugation of forms induces an isomorphism $$H^p(X,\Omega_X^q) \simeq \overline{H^q(X,\Omega_X^p)}$$
Let $...
- 604
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236
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bivariate polynomial
Hello,
Let $p(x,y) = \sum_{m=1}^M\sum_{n=1}^N a_{m,n}x^{m-1}y^{n-1}$ be a bivariate polynomial where $\{a_{m,n}\}$ are complex.
If $(x_k, y_k), k=1,2,\cdots, MN-1$ are roots of $p(x,y)=0$ where $|...
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88
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constant field of a divisor
Let $D$ be a divisor on a smooth, projective algebraic variety $X$ of dimension n over a field $k$ and let $D_j$ be an irreducible component of $D$.
I came across the expression the constant field ...
- 11
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158
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On the flatness of certain morphism
Assume $X, Z, S$ are shemes of finite type over $\mathbb{C}$, $X$ is also irreducible and reduced, $\phi: Z\to S$ is affine flat morphism with reduced connected fibers, $\psi: Z\to X$ is such that $\...
- 242
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479
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Globally generated subvector bundles and evaluation maps.
Let $E$ be a globally generated vector bundle on a surface $S$ of rank $r\geq 2$. By standard facts about degeneracy loci, for a general $V\in G(r,H^0(E))$ one has:
(*)the evaluation map $ev: V\...
- 763
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184
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If you perturb a polynomial by a smooth function, then is the signed number of small zeros of the perturbed equation the same as the lowest non zero derivative?
Let $f: \mathbb{C} \rightarrow \mathbb{C} $ be a function of the form
$$ f(z) = z^n + z^{n+ 1} g(z) $$
where $g$ is a $\textbf{smooth}$ function (not necessarily holomorphic).
Is it true that the ...
- 3,235
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114
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A certain 'coniveau-like' filtration for cohomology: what can one say about the intersection of $Ker H^i(X)\to H^i(Z)$ for $Z$ running through subvarieties of $X$ of dimension $m$?
Let $X$ be a smooth variety (say, a complex one; denote its dimension by $n$). What can one say about the intersection of $Ker (H^i(X)\to H^i(Z))$ for $Z$ running through (closed, not necessarily ...
- 16.5k
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Comparing the homogeneous defining ideals of multiple embeddings of a projective scheme
If $X$ is a projective scheme over a field $k$ (which we may assume is algebraically closed), then under an embedding $i: X \hookrightarrow \mathbb{P}^n_k$, we may write $X = Proj(R/I)$ where $R = k[...
- 11
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122
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Embedding of curves in surfaces
Let $C_1 \cup C_2$ be a curve in $\mathbb{P}^3$ and $X$ be a smooth degree $d$ surface in $\mathbb{P}^3$ containing them and $d \ge 6$. Further, assume that the minimum degree polynomial in $I(C_1 \...
- 1,020
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323
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System of polynomial equations: P(x)=P(y) rather than P(x)=0
$P$ is a system of polynomials in $n$ variables over $\mathbb{Q}$. $Q$ is a singe such polynomial. Let $V$ be the zeros of $Q$. I know from some symmetry argument that for every $y \in [0,1]^n \...
- 13
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212
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cohomological dimension, dimension of modules and arithmetic rank
Let $R$ be a noetherian ring and $I$ be an ideal of $R$ and let $M$ be a finitely generated $R$- module.
I know two fact. first, dimension of $M$(i.e. krull dimension of $R/{\rm ann}(M)$) is greater ...
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112
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Divisors, factorisations of matrix valued functions, and the Lorentz group
How to construct a complex projective variety with several classes of non-intersecting divisors? How to keep the answer concrete and simple, so that explicit calculations can be done? And the problem -...
- 1,213
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178
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relative normal crossing and normal base change
Hi,
I have the following situation
$ U\stackrel{i}{\rightarrow}X\stackrel{f}{\rightarrow} Y$
with $f$ a proper smooth morphism of schemes, $i:U\rightarrow X$ an open immersion such that $D=X\...
- 11
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256
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How does the line bundles look like on a proper model (or Néron model) of an abelian variety?
How does the line bundles look like on a proper model (or Néron model) of an abelian variety?
Who knows references about this?
In particular, let us work over a trait $S=\mathrm{Spec} R$, where $R$ ...
- 997
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190
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Special values of zeta functions and extensions of base fields.
Let $X$ be a scheme of finite type over a finite field $k=\mathbb{F}_{q}$ of $q$ elements.
Then, one can define the zeta function $Z_{X/k}(T)$ of $X$ ovet $k$ as $\prod_{x\in |X|}\frac{1}{1-T^{deg_{k}...
- 945
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138
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on a decomposition lemma in adelic groups
Let X a curve over an algebraically closed k.
Fix $x$ and $y$ two distinct closed points of X.
Let G be a connected reductive group over k.
We denote Spec $\hat{\mathcal{O}}_{X,x}$ the formal ...
- 3,432
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256
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Classical modular curves over characteristic $p$.
The classical modular curve $\Phi_n(X,Y) \in \mathbb{Z}[X,Y]$ for $n \in \mathbb{Z}_{\geq 2}$ relates the $j$-invariants of elliptic curves $E_1$ and $E_2$ defined over $\mathbb{C}$ in the sense that ...
- 163
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Does a (NOT necessarily positive) current have a decomposition formula?
It is well-known that for any positive (1,1)-current $T$, there is a decomposition formula according to [Siu74]. That is, $T$ can be written as an infinite sum of prime divisors plus an extra part. In ...
- 363
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How can one compute the cohomology of $i'^*C$, for $i':\mathbb{A}^{N-1}\setminus \{0\}\to \mathbb{A}^{N}\setminus \{0\} $?
For an (etale or 'topological', constructible bounded) complex of sheaves $C$ on $X'=\mathbb{A}^{N}\setminus \{0\} $, $i'$ being the embedding $\mathbb{A}^{N-1}\setminus \{0\}\to \mathbb{A}^{N}\...
- 16.5k
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269
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Stein factorization and thm. of formal functions
If $f: X \rightarrow Y$ is a proper morphism of locally noetherian schemes with $f_* \mathcal{O}_X = \mathcal{O}_Y$ then the thm. of formal functions tells us that $f$ has connected fibers, since ...
- 3,505
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177
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Certain vanishing of local cohomology supported on some exceptional divisor
Consider $V:= \mathbb{C}^n$ such that $n \ge 3$ and take $p \in V$. Let $\nu \colon \tilde{V} \rightarrow V$ be a composite of blow-ups along smooth centres such that $\nu$ is an isomorphism outside $...
- 909
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250
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Abelian variety
Let P(x,y, z, w)= 0 be a complex surface . How can we know that this surface is a sub-variety of an abelian variety or not?
- 99
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517
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Solving 3D equation system (inverse-projecting a triangle)
Please, how is the equation system below named exactly (to search further literature)?
Does it have an analytical solution? If it doesn't, then what could be the fastest numerical method for it (...
- 19
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192
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"Higher" Tangent spaces in char-p geometry - definition?
Hi, everyone!
I have some construction that requires exact definition.
I considered polynomial homomorphisms from $\Bbbk$ to $U_n(\Bbbk)$ (unitriangular matrices), where $char \Bbbk = p>0$ (more ...
- 1,412
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180
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Fields over which cubic hypersurfaces are rational
All cubic hypersurfaces having at least one double point are birational to some $P^n$ over an algebraically closed field. How does the statement change as I pass to non alg closed fields? Does it hold ...
- 3,717
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165
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How do Hodge classes for Calabi-Yau 4-folds compare with the classes for tori?
Let $X$ be a Calabi-Yau 4-fold, i.e., a connected 4-dimensional compact Kahler manifold with $K_{X} \cong \mathscr{O}_{X}$ and $h^{i} (X,O_{X} )= 0$ for $0 \lt i \lt 4$.
Given a general 4-...
- 119
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150
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Image of linear projection
Let $X \subset \mathbb{P}^n$ be a projective variety (i.e. zariski closed), and let $\pi : X \dashrightarrow \mathbb{P}^m$ a linear projection ($\pi$ is not in every point of $X$ defined).
Under ...
- 175
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231
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Set of Curves Passing through a smooth point of a Variety is Zariski-Dense
In a paper by F. Pop he claims the following fact-
Let $K$ be a field. The set (by which I believe he means the union) of all smooth $K$-curves passing through a smooth $K$-rational point of an ...
- 11
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238
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How Markus–Yamabe implies Jacobian ?
To make myself precise, I would like to recall some backgrounds.
(Markus-Yamabe, $\mathrm{MY}_n$) Given a $C^1$ map $f:\Bbb R^n \to \Bbb R^n$ with $f(0)=0$ and $Df$ everywhere Hurwitz stable (the ...
- 355
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153
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Trivialization of holomorphic symplectic 2-form
Let $X$ be a holomorphic symplectic compact manifold with a fixed holomorphic 2-form $\omega$. $\omega$ yields an isomorphism $\phi:T_{X} \rightarrow \Omega_{X}$ via
$$
v \mapsto \phi(v)=\omega(v,-).
...
- 583
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190
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dimension of spaces of rational curves in a variety!
Do you know calculate the dimension of the space of rational curves of degree
m, through d given points, contained in some projective variety?
- 63
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339
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Vanishing of cohomology groups
Given a smooth hypersurface $X \subset \mathbb{P}^{2p+1}_{\mathbb{C}}$ of degree $d \ge 5$ (with $p\ge 2$), when can we say that
$H^{p+1}(\mathcal{O}_X)=H^{p+1}(\Omega^1_X)=...=H^{p+1}(\Omega^{p-2}...
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