Questions tagged [ag.algebraic-geometry]
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
1,921
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Bounds on the coin-flipping degree
Let $p(\lambda)$ be a polynomial that maps the closed unit interval to itself and satisfies $0\lt p(\lambda)\lt 1$ whenever $0\lt\lambda\lt 1$.
The polynomial can be written in power form:
$$p(\lambda)...
- 637
2
votes
2
answers
362
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Does a generic curve inside the space of curves with a node at a specific point have only finitely many nodes?
Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of homogeneous degree $d$
polynomials in three variables (up to scaling), where $\delta_d = \frac{d(d+3)}{2}$.
Define $\mathcal{A}$ to be ...
- 3,235
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1
answer
1k
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Thom-Gysin Sequences and Stratifications
Let $X$ be an affine algebraic variety over $\mathbb{C}$, and let $G$ be a semisimple complex linear algebraic group acting by variety automorphisms with finitely many orbits. The decomposition of $X$ ...
- 4,870
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134
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Can logarithmic connection operate on currents?
Let $(X,\omega)$ be a compact Kähler manifold of dimension $n$, and let $D=\sum_{i=1}^r D_i$ be a simple normal crossing divisor on it, i.e., a divisor with smooth components $D_i$ intersecting each ...
- 325
2
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1
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455
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Deformations of pointed stable maps with "curve held rigid" or "preserving the dual graph"
I am reading the "Notes on stable maps and quantum cohomology" by Fulton and Pandharipande and I got stuck in the proof of the Theorem 2, p.27. The authors consider the space $Def(\mu)$ of first order ...
- 203
2
votes
2
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350
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Linkage between singularities of algebraic varieties and continued fractions
I have an impression that there is linkage or relation between singulariry of algebraic variety and continued fraction when I read some book on resolution of singularity or algebraic geometry.Could ...
- 3,000
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1
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197
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twisted cubic in a smooth hyperplane section of a cubic threefold
Let $X$ be a complex smooth cubic threefold and $C$ be a smooth twisted cubic, then $C\subset Y\subset X$ for a unique cubic surface $Y$ in $X$ (or equivalently a hyperplane section of $X$). When $Y$ ...
anon
2
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0
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254
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Chevalley groups over $k[t]/t^n$
This question is motivated partly by a recent question on Chevalley groups over arbitrary commutative rings (and see also this older question). The answers to that question point to a large and ...
- 3,627
2
votes
1
answer
271
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Minimum length path touching $n$ circles
Given $n$ non-overlapping circles of radius $1$ (i.e., the distance between the centers of any two circles is greater than $2$), how to find the minimum length path (the path can be of any form) that ...
- 436
2
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1
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166
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Global centralizers in Jordan-Chevalley decomposition in bad characteristic
Let $G$ be an affine algebraic group defined over an algebraically closed field $k$ of arbitrary characteristic, and write $\mathfrak{g}$ for its Lie algebra. Given $X\in\mathfrak{g}$, it has (...
- 53
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176
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Lifting a morphism along quotient of a group action
Let $X$ and $Y$ be complex projective varieties. Assume there is a finite group $G$ acting on $Y$ and we denote the quotient projective variety by $Y/G$. We have a morphism of $\mathcal{Hom}$-schemes ...
- 5,851
2
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1
answer
175
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Relations between Dolbeault cohomology and the corresponding $L^2$-cohomology
Let Dolbeault cohomology and the corresponding $L^2$-cohomology be denoted by $H^{p,q}(X) $
and $H^{p,q}_{(2)}(X)$ respectively.
As is well known, on a compact complex manifold $X$, $H^{p,...
- 391
2
votes
1
answer
200
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Intermediate extension and irreducible subquotients of perverse cohomologies
Let a set X be the union of two locally closed subsets U and V such that U does not lie in the closure of V. Let the restriction of a complex R of constructible sheaves on X to a smooth open subset A ...
- 2,490
2
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0
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388
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relative cohomology $H(X,D)$ of a pair in Weil cohomology theory
In algebraic topology, one defines relative cohomology groups $H(X,A)$ of a pair of spaces $A\subset X$.
Is there an analogue in algebraic geometry of cohomology of a pair of schemes?
For example, ...
- 1,299
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1
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227
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curve through a point avoiding an hypersurface, II
Inspired by this question:
Suppose given an algebraic curve $C \subset \mathbb{A}^2$, and a point $x \in C$. Can you find another (closed) curve $D \subset \mathbb{A}^2$ such that $C \cap D = x$?
- 8,643
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1
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481
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Stack associated to Groupoid object in category $\text{Sch}/S$
Consider the category of manifolds $\text{Man}$.
A groupoid object in the category of manifolds is called a Lie groupoid, denoted by $\mathcal{G}$. There is a way to associate a stack (over the ...
- 5,931
2
votes
1
answer
728
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Why is the Tate local duality pairing compatible with the Cartier duality pairing?
This question is a follow up to Why is the norm map dual to restriction under Tate local duality?
Let $A$ and $B$ be dual abelian schemes over a base scheme $S$. For an integer $n \ge 1$, consider ...
- 1,103
2
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1
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221
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How to complete $i^*i_*\mathcal{F}\to \mathcal{F}$ into an exact triangle for a smooth divisor $i: X\hookrightarrow Y$?
Let $Y$ be a smooth algebraic variety and $i: X\hookrightarrow Y$ be a smooth divisor. We consider the derived functors $i^*: D^b_{coh}(Y)\to D^b_{coh}(X)$ and $i_*: D^b_{coh}(X)\to D^b_{coh}(Y)$. By ...
- 8,707
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1
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542
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Proposition 1.5 in Mumford's Geometric Invariant Theory
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\pr{pr}$I have some problems to understand the proof of Proposition 1.5 from Mumford's ...
- 6,000
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1
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705
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A question on nested Hilbert scheme
Let $H_1, H_2$ be two Hilbert schemes parametrizing subschemes in $\mathbb{P}^{n_1}, \mathbb{P}^{n_2}$ with Hilbert polynomials $P_1, P_2$, respectively. Given a pair $(Z_1, Z_2)$ of subschemes in $\...
- 2,022
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2
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501
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spin bundle vs. hodge bundle
Let $\overline{\mathcal{M}}^r_{g,n}$ the space of $n$ pointed $r$-stable curves of genus $g$, endowed with a root of the canonical sheaf, and let $\mathcal{C} \to \overline{\mathcal{M}}^r_{g,n}$ be ...
- 3,717
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2
answers
1k
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Uniqueness on square root of complex Line Bundle
Let $L$ be a line bundle over a compex manifold $X$, a square-root of $L$ is a line bundle $M$ such that $M^{\otimes2}=L$. My question is when the square-root of Line Bundle is unique?
user21574
2
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3
answers
837
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Schemes with trivial brauer group
What are some nontrivial examples of schemes whose Brauer group is trivial? I am aware that rational varieties have trivial Brauer group and retract rational varieties in the sense of Saltman also do....
- 21
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70
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Does there exists a "local slice" for an action $ \widehat{\mathbb{G}_{a}} $ on $ \operatorname{Spf}(\widehat{A}) $ (char zero)?
Every action $ \beta $ of $ \mathbb{G}_{a} $ on a variety $ \operatorname{Spec}(A) $ over a field of characteristic zero is obtained from a locally nilpotent derivation $ \delta $ via $ f(t_{0} \ast x)...
- 789
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2
answers
338
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A natural bijection between the orbit spaces of $p$-nilpotent matrices for varying $p$
Let $k$ be an algebraically closed field of characteristic $p$, call a matrix $X\in\mathfrak{gl}_n(k)$ $p$-nilpotent if $X^p=0$, and let $\mathcal{N}_1=\mathcal{N}_1(\mathfrak{gl}_n(k))$ be the set of ...
- 768
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559
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commutative diagram with Yoneda pairing, Weil pairing and edge morphism
Why does the following diagram commute?$\require{AMScd}$
\begin{CD}
H^0(X,\mathscr{A}) \times \mathrm{Ext}^2_X(\mathscr{A},\mu_{\ell^n}) @>>> H^2(X,\mu_{\ell^n}) \\ @VVV @| \\
H^1(...
user19475
2
votes
2
answers
831
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Number of generators of an ideal in a polynomial ring over a Noetherian ring
Let $R$ be a Noetherian ring. By the Hilbert Basis Theorem the polynomial ring $R[x_1, \ldots , x_n]$ is also a Noetherian ring. What can we say about the number of generators of an ideal $I$ of $R[...
- 545
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1
answer
380
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Intersections of algebraic surfaces with hypercubes of a $d$-dimensional grid
This is a follow-up question, to a question I asked earlier.
See Algebraic curve intersecting square-grid.
Consider $n^d$ unit hypercubes in $d$-dimensional Euclidean space
tightly packed in the ...
- 469
2
votes
2
answers
994
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Injective resolution for right derived functor
This question is base on my previous question, and I repeat it here:
Suppose $X$ is a projective variety and $D^{+}(X)$ is the derived cateogry of bounded below complexes of sheaf of $\mathcal{O}_X$-...
- 3,362
2
votes
1
answer
740
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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
3
answers
910
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Are the arithmetic genera of Cohen-Macaulay curves in a fixed homology class bounded?
Let X be a smooth projective variety over the complex numbers. Recall that a Cohen-Macaulay curve is a one-dimensional closed subscheme without embedded or isolated points (fat components are allowed)....
- 2,198
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1
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1k
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Simple example of a perfect complex not isomorphic to a strictly perfect complex?
I'm looking for the simplest possible example (one that's easy to remember) for the situation described in the title. More precisely I'm looking for the following example:
A (probably has to be ...
- 7,549
2
votes
1
answer
511
views
Constraints on the base of an elliptically fibered Calabi-Yau threefold
Let $X\to B$ be an elliptic fibration over a base $B$. I assume that both $X$ and $B$ are smooth projective varieties. The elliptic fibration has a rational section.
If $X$ is a Calabi-Yau variety (...
- 2,982
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0
answers
257
views
Codimension restrictions on intersections
This is a question I stumbled across earlier this week. I see a similar one has been asked here.
Suppose I have a smooth quasi projective variety $X$ over a field $K$, and I call $\text{Chow}^r(X\...
user113452
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0
answers
158
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Interpretation of some maps involving cohomology groups
I've asked some questions on Math Stackexchange regarding areas around my research but I received very little success with responses, so I thought I might try to share some of my other problems here ...
- 865
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0
answers
325
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Irreducible surface singularity that is not a local set-theoretical complete intersection
I have been looking for a criterion for the germ of an irreducible complex surface singularity $(X,x)$ to be a set-theoretical complete intersection.
A germ $(X,x)$ of an isolated complex singularity ...
- 1,379
2
votes
1
answer
278
views
Can we have "tropical polynomials" with arbitrary real powers?
I am referring to the definition of a tropical polynomial given on page 8 (top of section 3) here in this review, https://arxiv.org/pdf/math/0306366.pdf. I understand that here the notion of a ...
- 2,146
2
votes
3
answers
318
views
Linear homogenous polynomials that generates one quadratic polynomial
Let $P_1, \ldots, P_m$, $Q_1, \ldots, Q_k \in \mathbb{C}[x_0,\ldots,x_n]$ be linear homogenous polynomials. Let $f$ be a homogenous quadratic polynomial of degree $2$.
Assume that for every $i$ and ...
- 2,683
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0
answers
132
views
Chebyshev-like Problem for Plucker Coordinates
$\newcommand{\R}{\mathbb{R}}\newcommand{\D}[1]{\Delta_{#1}}\newcommand{\set}[1]{\{#1\}}\newcommand{\abs}[1]{\lvert#1\rvert}$
Let $n=2d+1$ be an odd integer, let $Gr(2,n)$ denote the Grassmmanian over $...
- 181
2
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1
answer
362
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Uniform position principle
The uniform position theorem says that the points of a general hyperplane section of an irreducible curve lie in uniform position. Doesn't that imply that the points of a general subspace of ...
- 325
2
votes
1
answer
287
views
Lüroth theorem for $k \subset k(f,g) \subseteq k(x)$
Let $k$ be a field of characteristic zero (I do not mind to assume that $k=\mathbb{C}$, if things are easier in this case).
Lüroth theorem says that a field $L$, $k \subset L \subset k(x)$ containing ...
- 2,783
2
votes
1
answer
357
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(Bridgeland stability conditions) How can I get the heart of a bounded t-structure on $D^b(P^3)$?
In the article, Bayer, Arend; Macrì, Emanuele; Toda, Yukinobu, Bridgeland stability conditions on threefolds. I Bogomolov-Gieseker type inequalities, J. Algebr. Geom. 23, No. 1, 117-163 (2014). ...
- 21
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0
answers
204
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Sylvester-Gallai-type theorem for quadratic polynomials
Let $F_1, F_2$ and $F_3$ be finites sets of irreducible polynomials in $\mathbb{C}[x_0, \ldots, x_n]$ of degree at most $2$ such that $F_1 \cap F_2 \cap F_3 = \varnothing$ and for every $Q_1, Q_2$ ...
- 2,683
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votes
1
answer
183
views
Is every sufficiently general monic quartic rational square infinitely often?
Let $f(x)=x^4+b_3 x^3+ b_2 x^2+b_1 x + b_0$.
Let $g(x)=x^4 f(1/x)$. Let $C : g(x)=y^2$.
$C$ is birationally equivalent to $f(x)=y^2$.
The constant coefficient of $g(x)$ is $1$ since $f$ is monic
and $(...
- 24.2k
2
votes
1
answer
583
views
Are finite correspondances flat?
In Voevodsky&Mazza&Weibel's book on motivic cohomology they define "an elementary correspondance from X (Smooth connected scheme over $k$) to Y (separated scheme over $k$) as an irreducible ...
- 2,741
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votes
1
answer
487
views
extending homomorphisms of Abelian schemes
Let $S$ be an integral scheme with function field $K = K(S)$. Let $\mathscr{A}, \mathscr{B}$ be Abelian schemes over $S$. Let $L/K$ be a separable field extension. Given $f_L \in \mathrm{Hom}(\mathscr{...
user19475
2
votes
1
answer
432
views
Strict transform under resolution of singularity along a singular $\mathbb{Q}$-Cartier divisor
$\DeclareMathOperator\Bl{Bl}$Let $f: Y=\Bl_0^\omega(\mathbb{C}^3)\to \mathbb{C}^3$ be a weighted blow up of $\mathbb{C}^3$ with weights $w(x,y,z)=(1,1,2)$. Then $Y$ and the exceptional divisor $E\cong ...
- 41
2
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0
answers
136
views
Covering a space by cones
Let $X\subset\mathbb{R}^n$ be some connected and bounded $n$-dimensional manifold, e.g. a space homeomorphic to an open/closed ball with possibly some parts of it being removed.
I am interested in ...
- 3,071
2
votes
1
answer
420
views
Classifying spaces and Brown's representability theorem
Let $G\text{-}PF(X)$ be the set of isomorphism classes of principal topological fibrations over the space $X$ with structural group $G$, and $G\text{-}PF_{cw} : hCW \to Set$ the contravariant functor $...
- 1,326
2
votes
0
answers
127
views
Coordinate ring of an equivariant embedding of a homogeneous projective variety
Lie algebra: Let $G$ be a semisimple, simply connected linear algebraic group with a fixed Borel subgroup $B$. Let $P$ be a parabolic subgroup containing $B$. Let $\lambda$ be a dominant integral ...
- 187