Questions tagged [ag.algebraic-geometry]
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
21,608
questions
2
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On the generating series of degree $d>1$ Gromov-Witten invariants of the local $\mathbb P^1$
Let $N$ be the total space of the vector bundle $\mathscr O_{\mathbb P^1}(-1)\oplus \mathscr O_{\mathbb P^1}(-1)$ over $\mathbb P^1$, and let $C_0\subset N$ be the zero section. Then $N$ is a quasi-...
- 1,534
2
votes
1
answer
227
views
The stable general linear group in algebraic geometry
Let $A$ be a $ \mathbb{C}$-algebra. Then we have the following increasing union:
$$ {\rm GL}_1(A) \subseteq {\rm GL}_2(A) \subseteq {\rm GL}_3(A) \subseteq \dots \subset {\rm GL}(A) $$
We call $ {\rm ...
- 3,710
2
votes
1
answer
387
views
Generic vs General property of reducedness in a family of projective schemes
Let $\pi:\mathcal{X} \to B$ be a flat family of projective schemes, $B$ is irreducible. Let $\mathrm{Spec} K$ be a generic point on $B$. Denote by $\mathcal{X}_K$, the pull-back of $\mathcal{X}$. This ...
- 823
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votes
1
answer
523
views
Castelnuovo Mumford Regularity
Consider the polynomial ring $S = C[x_1,\ldots,x_n]$. Let
$$
0 \rightarrow E_{n-1} \rightarrow \cdots \rightarrow E_1 \rightarrow E_0 \rightarrow I \rightarrow 0 ,
$$
be a ...
- 73
2
votes
1
answer
355
views
Moduli of curves in characteristic zero
Let $K$ be a field of characteristic zero, and let $\overline{K}$ be its algebraic closure. Let $\overline{M}_{g,n}(K)$ and $\overline{M}_{g,n}(\overline{K})$ be the coarse moduli spaces parametrizing ...
user55899
2
votes
1
answer
297
views
Two questions about counter example for Torelli theorem for hyperkahler manifolds
i am reading the article "Counter-example to global Torelli problem for irreducible symplectic manifolds" by Yoshinori Namikawa and i have two questions i can't answer:
1) he takes $T$ a complex ...
- 23
2
votes
1
answer
607
views
The stability of vector bundle with trivial Chern classes is independent of ample divisor, a direct proof?
Let $X$ be a smooth projective variety over $\mathbb{C}$. For an ample divisor H, we can define the slop of vector bundle with respect to $H$, then we can define stablilty of vector bundle with ...
- 699
2
votes
1
answer
457
views
examples of Kähler manifolds with trivial Hodge numbers and first Chern classes
Yesterday I asked the following question to which abx has given a positive answer.
examples of Kähler manifolds with trivial odd Betti numbers and first Chern classes
But I suddenly realized ...
- 483
2
votes
1
answer
179
views
The target of a regular function in Non-archimedean analytic geometry
Let $(k,|\cdot|)$ be an algebraically closed field, complete wrt a (multiplicative) norm as in the framework of the Berkovich's analytic geometry. Given a commutative Banach $k$-algebra $\mathcal{A}\...
- 375
2
votes
1
answer
1k
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intersection pairing and cup product
Let $X$ be a smooth quasi-projective algebraic variety over $\mathbb{C}$ and $A^k(X)$ be the Chow group of codimension-$k$ algebraic cycles on $X$. let $\mathrm{cl}$ be the cycle map from $A^k(X)$ ...
- 273
2
votes
1
answer
740
views
Canonical lifts from $\mathbb F_q$ and CM-theory
One knows that (ordinary) Jacobians of hyperelliptic curves over a finite field $\mathbb F_q$ (mostly of genus 1 (elliptic curves) and 2) are extensively studied by cryptographers, as a platform for ...
- 637
2
votes
1
answer
806
views
Properties of extreme rays
Let $X$ be a projective variety over $\mathbb{C}$, then the effective curves module the numerical equivalence form a cone. For my understanding, if the extreme ray $[C]$ has the property that $[C]\...
- 3,362
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votes
1
answer
180
views
When is the Clifford index of a curve computed by pencils?
Under which circumstances is the Clifford index of a curve computed by pencils?
- 761
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1
answer
379
views
Root discriminant lower bounds in algebraic geometry
Let $X$ be a simply-connected smooth projective variety over $\mathbb C$. Let $C$ be a curve on $X$.
If $Y$ is a ramified cover of $X$ of degree $n$, and $D$ is the branch divisor of $Y$, call $(D \...
- 137k
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1
answer
318
views
A functorial property of higher right derived functors
Let $f:X \to Y$ be a projective morphism of complex Noetherian schemes. Assume $Y$ is smooth and for all $y \in Y$, $f^{-1}(y)$ is of pure dimension $1$. Let $\mathcal{F}_1, \mathcal{F}_2$ and $\...
- 823
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votes
1
answer
323
views
Locally trivial deformations of surfaces with quotient singularities
Let us consider the surface $\mathbb{A}^{2}/\mu_{6}$ where the action is given by
$$
\begin{array}{ccc}
\mu_{6}\times\mathbb{A}^{2} & \longrightarrow & \mathbb{A}^{2}\\
(\epsilon,x_{1},x_{2}) ...
- 8,852
2
votes
1
answer
737
views
Does the normalization morphism induce isomorphism on residue fields?
The question is basically coming from the following situation:
Let $C$ be an integral curve over a field $k$ (EDIT and assume that $k$ is not algebraically closed) and let $\phi\colon C^N\to C$ be the ...
- 165
2
votes
1
answer
140
views
Depth of Schubert cycles
For $a:a_1\geq \cdots\geq a_c$, let $\sigma_a$ be the corresponding Schubert cycle over $Gr(c,\infty)$. We say $a$ is of depth $k$ if $a_1-a_c=k$ ($c>1$). Let $a$ and $b$ be of depth $k_1$ and $k_2$...
2
votes
1
answer
262
views
Hilbert polynomial of a projected variety
If $X$ is a variety and $X_p$ is its projection from a point $p$ not on $X$, can we relate the Hilbert polynomial of $X_p$ to the one of $X$?
- 325
2
votes
1
answer
429
views
Explicit basis for the space of global sections of a twisted arithmetic ideal sheaf
Assume $x\in X=\mathbb{P}^1_{\mathbb{Z}}$ is a closed point with $f(x)=p\in Y$ where $f:X\rightarrow Y$, here $Y=Spec(\mathbb{Z})$. Assume $k(x)=\mathbb{F}_p$ and denote by $I_x$ the ideal sheaf of $x$...
- 251
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1
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163
views
stable vector bundle and space surves
I am sure this is well known, but I am not an expert...so I appreciate any help
Let $C \subset \mathbb{P}^3$ be the complete intersection of two hypersurfaces of degree $d_1$ and $d_2$. Let $U_{d_1,...
- 21
2
votes
1
answer
466
views
descent in derived category
Suppose we are given a field $k$ and a finite Galois extension $L$ with Galois group $G$. We consider the projection $\pi:X\otimes_k L\rightarrow X$ for a smooth projective variety $X$. The object $\...
- 731
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votes
1
answer
522
views
Geometric interpretation of a (standard) commutative algebra fact
Which is your geometric interpretation (if any) of the following commutative algebra proposition?
Proposition. Let $M$ be a finitely generated $A$-module, $I\subseteq A$ an ideal, and $\phi\in \...
- 22.7k
2
votes
1
answer
430
views
Resolution of singularity of polynomials
Let $f$ be polynomial on a vector space $V$. Let $Z$ be the zero set of $f$ in $V$. Let $Z_{sing}$ be the singular part of $Z$.
By Hironaka's desingularization theorem, there exists a birational map ...
- 1,447
2
votes
1
answer
646
views
Kodaira dimension of the moduli space of curves
It is known that the moduli space $\overline{M}_{g}$ of genus $g$ curves is of general type for $g\geq 24$.
By Theorem 2.4 of
Logan, Adam The Kodaira dimension of moduli spaces of curves with ...
- 8,852
2
votes
1
answer
816
views
Moduli space of stable principal $G$-bundles
We have this Mumford's theorem:
Let $X$ be a Riemann surface of genus $g$, $G$ a simple Lie group. We can consider a principal stable $G$-bundles over $X$ (say $\xi$), where $rk(\xi)=r$ and $deg(\xi)=...
- 243
2
votes
1
answer
285
views
F-curves and divisors on $\overline{\mathcal{M}}_{g,n}$
It is conjectured that a divisor on $\overline{\mathcal{M}}_{g,n}$ would be ample if $D\cdot C >0$ for all F-curves $C\subset \overline{\mathcal{M}}_{g,n}$. Does the intersection degree with all F-...
- 3,717
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votes
1
answer
723
views
Do versions of the Nakai-Moishezon and Kleiman criteria hold for Moishezon manifolds, or other 'nice' spaces?
As I understand it, the Nakai-Moishezon criterion gives conditions for the existence of an ample divisor class on an arbitrary proper scheme, and Kleiman's criterion does the same for arbitrary ...
- 884
2
votes
1
answer
388
views
Difference between Kahler-Einstein and Bergman metric on a bounded symmetric domain
Let $H$ be a bounded symmetric domain.
What is the difference between the Bergman metric and the Kahler-Einstein metric on $H$?
- 103
2
votes
1
answer
252
views
On Non F-pure ideal and Sharp F-Purity for a pair $(X, \Delta)$ where $K_X+\Delta$ is NOT $\mathbb{Q}$-Cartier
Suppose $(X,\Delta\ge 0)$ is a pair such that $(p^g-1)(K_X+\Delta)$ is an Integral Weil Divisor for some $g>0$ and $X$ is a normal variety. Define $\mathcal{L}_{e,\Delta} = \mathcal{O}_X( (1-p^g)(...
- 165
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votes
1
answer
899
views
Paralel bezier curve
If I have a cubic Bezier curve specified by two endpoints and two control points, how can I find an offset curve which is "parallel" to the original at some given distance, after i have determined the ...
- 123
2
votes
1
answer
332
views
on the zariski closure and cocompactness of discrete groups
Assume $\Gamma$ is a discrete subgroup of some $GL_n$, and let $G$ be its Zariski closure.
Let $H$ an algebraic cocompact normal subgroup of $G$. Do we have that $H\cap \Gamma$ is of finite index in ...
- 23
2
votes
1
answer
417
views
Local blowup versus global blowup
Let $R=k[x_1,..,x_n]/I$ and let $X=Spec(R)$ be it's associated affine scheme. Suppose that $X$ has only one isolated singularity, say at the origin $\mathfrak{m}=\langle x_1,...,x_n\rangle$. Now, ...
- 145
2
votes
1
answer
137
views
Embeddings of of quotient singularities
Let $G$ be a finite subgroup of $SL(n,\mathbb{C})$. Let $G$ act on $\mathbb{C}^{n}$ with the action induced by $SL(n,\mathbb{C})$ and the induced action of $G$ on the unit sphere $S^{2n-1}$ is free. ...
- 1,707
2
votes
1
answer
155
views
compact support cohomology and finite morphism
Let X be a smooth complex algebraic variety and $\pi: X \to Y$ a finite morphism. Is it true
that
$H_c^k(X, \mathbb{Q})=H_c^k(Y, \pi_\ast\mathbb{Q})$?
Here $H_c$ stands por cohomology with compact ...
2
votes
1
answer
235
views
Two lines with orthogonal Plucker embedding
Let $l_1$ and $l_2$ be two lines in $G(1,n)$, the Grassmannian of lines in n dimensional projective space.
Suppose that their Plucker embeddings has dot product zero. Namely if $(x_1, x_2, \cdots, x_N)...
- 503
2
votes
2
answers
384
views
smooth modular compactification of moduli of curves
Is there a smooth modular compactification of the moduli space of smooth curves of genus $ g > 1 $ over $ \mathbb{C} $?
I am willing to allow for enrichments such as level structures. The ...
2
votes
1
answer
545
views
equivariant Serre Duality.
Let $X$ be a nonsingular projective variety of dimension $n$ over a field $k$, and $\omega_X$ be its canonical sheaf. Let $G$ be a finite subgroup of the automorphism group $Aut_k(X)$, and $\mathcal{...
- 570
2
votes
1
answer
1k
views
Skyscraper sheaf under Fourier-Mukai transform
Let $X,Y$ be smooth varieties defined over $k$. Suppose $P$ is a coherent sheaf on $X \times Y$ flat over $X$, considering the Fourier-Mukai transform
$$\Phi_P : D^{b}(X) \to D^{b}(Y)$$
which is ...
- 3,362
2
votes
1
answer
296
views
Lifting vector fields to its resolution in char $p$
In this paper 4.7, the authors showed that on a normal variety $X$, if there is a tangent vector field on its smooth locus, then it can be lifted as a logarithmic tangent vector field on a log ...
- 656
2
votes
1
answer
442
views
Crepant Birational Map on the Blow-up
Let $ f: \mathbb{P}^n \dashrightarrow (\mathbb{P}^1)^n , [x_0:\dots :x_n] \mapsto ([x_0,x_1],[x_0,x_2], \dots ,[x_0,x_n])$ a birational map.
In particular, if $X$ is the blow-up of $\mathbb{P}^n$ at $...
- 325
2
votes
2
answers
679
views
Normal bundle of a component of a reducible fiber
Suppose that $\pi : X \to S$ is an elliptic fibration, and that the fiber over a point $s \in S$ is reducible, with several components. Call one of them $C$. How can I compute the normal bundle of $C$...
- 51
2
votes
2
answers
655
views
Leray spectral sequence of the inclusion of an open subvariety
Let $X$ be a smooth variety over a field $k \subset \mathbb{C}$ and $Z$ a smooth subvariety. Let $U=X-Z$. I'm trying to understand what information do the Leray spectral sequences attached to the ...
- 23
2
votes
3
answers
188
views
How to tell if a second-order curve goes below the $x$ axis?
Suppose we have a second-order curve in general form:
(1) $a_{11}x^{2}+2a_{12}xy+a_{22}y^{2}+2a_{13}x+2a_{23}y+a_{33}=0$.
I'd like to know if there is a simple condition that ensures that the curve ...
- 6,882
2
votes
1
answer
579
views
Picture of a 3 dimensional amoeba.
On Wikipedia there some pictures of two dimensional amoebas (Thanks to Oleg Alexandrov for the pictures and the Matlab code he gives to build them). I was wondering if somewhere there are pictures ...
- 183
2
votes
1
answer
237
views
For which local $R$ its K-theory mod l is isomorphic to the one of its residue field?
It is well-known (and was proved by Gabber?): if $R$ is a regular henselian local ring containing a field of characteristic prime to $l$, $k$ is its residue field, then $K_\ast(R,\mathbb{Z}/l)\cong ...
- 16.5k
2
votes
1
answer
338
views
Example of special Lagrangian fibration of compact CY3?
I would like to know an explicit example of a special Lagrangian fibration of compact Calabi-Yau 3-folds. Are there any example known among experts? I know that there are some for noncompact Calabi-...
- 23
2
votes
1
answer
496
views
divisors and powers of line bundles
Can anyone help me with the following question? Let $X$ be a smooth, projective algebraic variety over a field $k$ of characteristic zero. Let $D$ be an effective divisor on $X$ and $m \geq 2$ an ...
- 23
2
votes
1
answer
554
views
Tannakian fundamental group for finitely linear representation of group
Let $G$ be an arbitrary group and $k$ a field. Denote by $Rep_{k}(G)$ the category of finite dimensional representations of $G$ over $k$. The usual tensor product and dual operations for ...
- 1,911
2
votes
1
answer
200
views
recover trace of l-adic sheaves defined over an extension
Let $X$ be an (as nice as you prefer) alg. variety (or alg. stack) defined over $\mathbb{F}_{q}$ and let $\mathcal{F}$ be an l-adic sheaf on $X_n = X {\times_{\mathbb{F}q}} \mathbb{F}_{q^n}$. Fix an ...
- 867