All Questions
22,770 questions
0
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51
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Does this tangent developable-like surface have a cusp along a curve or is it smooth?
Consider the following surface $X$ which is a subvariety of the full flag variety $Y=\{0 \subset V_1 \subset V_2 \subset V\}$ where $V$ is a fixed three-dimensional vector space and ${\rm dim} V_i =i$....
- 2,964
1
vote
1
answer
81
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Vector bundle formed by tangent lines to a quadric curve in $\mathbb P^2$
Let $Q \cong \mathbb P^1$ be a quadric curve in $\mathbb P^2$. Consider the following rank two vector bundle $V$ on $Q$: the fiber of $V$ over a point $p$ of $Q$ is the two-dimensional subspace $V_p$ ...
- 2,964
3
votes
1
answer
180
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How to find equations of $\mathbb{C}^*$-curves
Fix positive integers $t_1,t_2,t_3$.
Suppose we have a $\mathbb{C}^*$ action on $\mathbb{C}^3\setminus\{0\}$ defined by
$$\mathbb{C}^* \times \mathbb{C}^3 \setminus \{0\} \to \mathbb{C}^3 \setminus \{...
- 1,981
0
votes
1
answer
113
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$\mathbb P^1$-bundle on a partial flag variety
Let $X$ be the partial flag variety of flags $0 \subset V_k \subset V_{k+2} \subset V$ where $V$ is a fixed vector space of dimension $n$ and ${\rm dim} V_k = k$ and ${\rm dim} V_{k+2} = k+2$. Is it ...
- 2,964
1
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0
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62
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Geometric stability conditions on calabi-yau's fibred over Fano always identical to geometric stability conditions on Fano
I apologize in advance for the long title. This question is motivated primarily by [2], with the explicit example of $\mathbb{P}^2$ and $\omega_{\mathbb{P}^2}$ computed in [3] and [1], respectively.
...
- 317
2
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0
answers
120
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Looking at versions of Implicit Function Theorem (IFT) on rings
$ \let \ovr \overline
\def \Z {\mathbb Z}
\def \C {\mathbb C}
\def \F {\mathbb F}
\def \P {\mathcal P}
\def \x {\boldsymbol x}
\def \a {\boldsymbol a} $
Let $ \P = \{ p _ i ( \x , y ) \} _ { i = 1 } ^ ...
- 346
4
votes
1
answer
271
views
Is there a non-semistable simple sheaf?
Let $C$ be a smooth projective (irreducible) curve over an algebraically closed field $k$.
A sheaf is said to be simple if its endomorphism algebra is isomorphic to $k$.
It is known that a stable ...
- 405
1
vote
0
answers
76
views
An upper bound of order of prime divisors
Let $X$ be a normal projective variety, $Z\subseteq X$ a closed subvariety, and suppose that $\mathfrak{a}\subseteq \mathcal{O}_X$ is an ideal sheaf. I tried to prove the following:
There is a ...
- 11
2
votes
1
answer
193
views
Global sections of tangent sheaf of singular varieties
Let $X\subset \mathbf{P}^{n+1}$ be a $n$-dimensional normal hypersurface of degree $3$, and we denote its tangent sheaf by $T_X$. We further assume that $n\geq 4$.
When $X$ is smooth, it is known that ...
- 389
1
vote
0
answers
124
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Section of étale morphism of algebraic spaces
I am sorry in advance if this question is too naive for specialists. I just realized that I need it when doin research and I haven't taken any serious course on algebraic spaces. Let $u \colon U \...
- 883
8
votes
0
answers
286
views
What is the current research situation of the Cheeger–Goresky–MacPherson conjecture?
In [CGM-1983], J. Cheeger, M. Goresky and R. MacPherson conjectured that the intersection cohomology of a singular complex projective algebraic variety $X$ is naturally isomorphic to its $L^2$-
...
4
votes
1
answer
321
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Cohomological range of a perverse sheaf
I want to prove that, let $X$ be a smooth variety, then for a local system $L$ on $X$, $L[\dim X]$ has no quotient object in the abelian category of perverse sheaves supported on a proper closed ...
- 1,168
1
vote
1
answer
210
views
The Étale Cohomology from the Variety to its Generic Point
Let $n\in \mathbb{Z}_{>0}$ and $X$ be a smooth projective variety over $k$, where $k$ contains all $n$-th roots of unity and $\operatorname{char}(k)=p$. Here $p$ and $n$ are coprime. I wonder ...
- 13
5
votes
1
answer
151
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Dimension from Hilbert series with variable-weighted grading?
Suppose $R = F[x_1,...,x_n]/I$ is a finitely generated ring over the field $F$, and suppose there is a function $d:[n] \to \mathbb{Z}$ such that defining the degree of a monomial $x_1^{e_1} ... x_n^{...
- 3,230
1
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0
answers
104
views
Is a normal domain a filtered colimit of Noetherian normal domains?
As described in the title, is any normal domain a filtered colimit of Noetherian normal domains? It will be great if one can show this, even with additional conditions, or if one can provide a ...
- 1,024
2
votes
1
answer
132
views
What is the polarization type of the push-forward of the Poincaré-bundle to the Jacobian of a curve?
$\DeclareMathOperator{\Jac}{Jac}\DeclareMathOperator{\Pic}{Pic}$Let $C$ be a smooth curve of genus $g > 0$, and consider the Picard torus $\Pic^d(C)$ of line bundles of degree $d$. Let $\mathcal P$ ...
- 1,286
4
votes
0
answers
95
views
Formula for bound on number of smooth projective toric Fano varieties of dimension n
In dimension 1, the only smooth projective toric Fano variety is $\mathbb{P}^1$. In dimension $2$, there are 5: $\mathbb{P}^1\times \mathbb{P}^1$, and then successive blow-ups of $\mathbb{P}^2$ at up ...
- 511
3
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0
answers
220
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Computing pushforwards and pullbacks of D-modules
Let $X$ be a smooth algebraic variety (over some field of char 0), $Z$ a smooth closed subvariety of codimension 1, $i : Z \hookrightarrow X$ the inclusion, and $j : U \hookrightarrow X$ the ...
- 37k
2
votes
1
answer
326
views
Extension by zero operation
Suppose you have a closed subset $Z$ of a topological space $X$, and $F$ is a sheaf on $Z$. Then one can consider the extension by zero sheaf $F^X$ on $X$.
What are some examples and situations which ...
- 129
1
vote
0
answers
124
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Space of all orthogonal partially complex $2\times2\times2$ tensors
I am trying to understand the space of all orthogonal tensors, I asked a more general version of this question here but with no solution yet found. I want to look at the simplest case first, namely a $...
- 101
2
votes
0
answers
126
views
Relative Jacobian as a ramified holomorphic quotient
Let $f:X \to S$ be an elliptic fibration with only $m$ singular fibers of type $I_1$ at the set of points $\lbrace s_1,\cdots, s_m \rbrace$ of $S$. In the paper "On Compact Analytic Surfaces: II&...
- 41
2
votes
1
answer
149
views
Flexes and projective equivalence of smooth cubics
I am trying to study Kock and Vainsencher's book "An invitation to Quantum Cohomology", working my way through the exercises. One of them ($0$th chapter) asks two prove that two elliptic ...
7
votes
1
answer
438
views
Road map and references for combinatorial Hodge theory
I'm a PhD student. I'm familiar with graduate level algebraic geometry and toric varieties.
I wanted to know a road map for getting into combinatorial Hodge theory and other prerequisites that I'll ...
- 839
2
votes
0
answers
146
views
Two open contractible subset of $\mathbb{R}^3$ which are not homeomorphic but are polynomial preimage of open sets of $\mathbb{R}$
About 2 decades ago I heared from some one that there are infinitely many open contractible subsets of space mutually non homeomorphic to each other. I confess that I do not remember the ...
- 356
2
votes
1
answer
154
views
Extending line bundle to regular model
Let $S$ be the spectrum of an excellent discrete valuation ring with field of fractions $K$ and $C$ be a proper integral regular curve over $K$. Assume, $C$ admits a proper regular flat model $\...
- 5,998
2
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0
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94
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Crepant resolution of cyclic quotient of affine space
Let $ G $ be a cyclic group of order $ n $, acting on $ \mathbb{C}^n $ by the cyclic action $ (z_1, z_2, \ldots, z_n) \rightarrow (z_2, z_3, \ldots, z_1) $. Does the quotient $ \mathbb{C}^n / G $ (...
- 936
1
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0
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143
views
Existence and injectivity of a map from $\mathrm{Num}(X) \otimes_\mathbb{Z} \mathbb{C}$ to $H^1(X,\Omega^1_X)$
I am currently reading a paper by Mustata and Popa on the Van de Ven theorem, you can read the paper here.
On page 51 there is the following map
$$\alpha_\mathbb{C} : \mathrm{Num}(X) \otimes_\mathbb{Z}...
- 101
2
votes
0
answers
104
views
Is the paracanonical bundle very ample?
Let $C$ be a generic curve (over algebraically closed field of characteristic $0$) of genus $g\geq10$ and $\eta$ a non-trivial torsion line bundle of level $l\geq3$ i.e. $\eta^{\otimes l}\cong\mathcal{...
- 439
3
votes
0
answers
111
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A non-normal scheme with infinitely generated Picard group
It's one of these standard facts that the Picard group of a normal scheme of finite type over $\mathbb{Q}$, or, more generally, an absolutely finitely generated field of characteristic $0$, is ...
- 544
1
vote
0
answers
98
views
Points on a rigid analytic variety and "points" on a formal model
Let $k$ be a finite extension of $\mathbb{Q}_p$. Let $X$ be a quasi-compact, quasi-separated rigid analytic variety over $k$. We choose a formal model $\mathcal{X}$ of $X$ over $\mathcal{O}_k$.
If I ...
- 519
6
votes
1
answer
822
views
Chromatic homotopy + algebraic geometry =?
In Homotopy Theory there is a famous theorem which shows that every cohomology theory satisfying a certain list of axioms is characterized by a formal group law, and that the spectrum associated to ...
- 2,907
2
votes
0
answers
118
views
Growth of invariants in mod-$p$ representations of $\mathrm{GL}_n$
Let $G$ be smooth admissible mod-$p$ representations of $\mathrm{GL}_n(\mathbb{Q}_p)$. Also suppose $\pi$ is an irreducible infinite-dimensional smooth admissible representation of $G$ over $\mathbb{F}...
0
votes
2
answers
320
views
K3 surfaces and density of rational curves
A smooth, complex, projective surface, such that the canonical bundle is trivial and the irregularity is equal to zero is called a K3 surface. Recently I received feedback regarding work I had done. ...
- 912
5
votes
1
answer
338
views
Equations for dual cubic curves
Suppose I have a cubic curve $C$ (over $\mathbb C$) in Weierstrass form $$y^2=x^3+ax+b.$$
I would like to find the degree $6$ equation for protectively dual curve $C^*$. Do you know any place where ...
- 14.3k
3
votes
1
answer
130
views
On Weil's theorem that a rational group action becomes regular action after some birational modification
People attribute the following theorem to Weil:
Any variety $X$ equipped with a birational action of a connected algebraic group $G$ is equivariantly birationally isomorphic to a variety $Y$ equipped ...
- 3,472
2
votes
0
answers
171
views
A conjecture on the scheme-theoretic image of a moduli map
Let $K/\mathbb{Q}_p$ be a finite extension with residue field $k$, and let $K'/K$ be a finite tamely ramified Galois extension with residue field $k'$. Let $E/\mathbb{Q}_p$ be a sufficiently large ...
5
votes
0
answers
144
views
Why do monoidal functors between categories of quasicoherent sheaves commute with external tensor products
I'm reading Lurie's paper on Tannaka duality for geometric stacks. Very roughly, my question is, why do monoidal functors, from which we try to build geometric morphisms, commute with certain algebra ...
- 1,374
1
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0
answers
216
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Dimension under change of ground field
I apologize if this question is "too elementary" - I am not an algebraic geometer. Are the following statements true?
Let $k\subset K$ an extension of algebraically closed fields of ...
- 19
3
votes
1
answer
270
views
degeneration of a Veronese surface
Let $V$ be the Veronese surface, obtained as the image of $\mathbb{P}^2$ in $\mathbb{P}^5$ by the complete linear system of conics. I understand that $V$ can degenerate to the union of a cubic scroll $...
- 3,779
4
votes
0
answers
178
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Splitting of counit-trace map for $\ell$-adic sheaf $\Bbb Q_{\ell}$
I have a question about following argument on page 3 in paper arxiv.org/abs/1702.04404 by Will Sawin (Proposition 3):
The claim is that for a finite, dominant map $f:X \to Y$ between varieties $X,Y$ ...
- 5,998
5
votes
1
answer
327
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Comparison between pushforward-pullback and quasi-coherent pushforward-pullback
In the following, an algebraic stack means a stack over the big fppf site of a scheme, admitting a smooth, representable, surjective morphism from a scheme (no separation hypotheses), and let $\mathsf{...
- 1,349
0
votes
1
answer
178
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Cohomology of Blow-ups and Minimal Models in Higher Dimensions
Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic zero. Consider a sequence of blow-ups:
$$X_n \xrightarrow{\pi_n} X_{n-1} \xrightarrow{\pi_{n-1}} \cdots \...
- 17
10
votes
2
answers
286
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Finding a model for $X_G$ with $G\subseteq \mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})$
I'm studying modular curves as part of my doctorate work and would like to understand how one gets from a subgroup $G$ of $\operatorname{GL}_2(\mathbb{Z}/N\mathbb{Z})$ to an equation for $X_G$. By $...
- 195
0
votes
0
answers
114
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Clarifications sought on the paper on the semigroup associated with a free polynomial by Ali Abbas and Abdallah Assi
I have three questions regarding the proof of Proposition 4 on page 4 of this paper here. For those interested in addressing these questions, please refer to some definitions in the first two or three ...
1
vote
0
answers
177
views
Prop 1.3 in "Birational geometry of algebraic varieties": specialization of rational curves
I have a couple of questions about some arguments in proof of Proposition 1.3 from Birational geometry of algebraic varieties by Kollár and Mori (p 8):
Proposition 1.3. [Abh56, Prop. 4] Let $X$ be ...
- 5,998
1
vote
0
answers
88
views
Identification of different components of Hilbert modular surface?
I'm wondering whether the different components of the Hilbert modular surface can be (naturally?) identified with each other, or if they're at least abstractly isomorphic. (I'd also be interested in ...
- 2,044
2
votes
1
answer
68
views
When are solutions to a set of multi-variate quadratic equations isolated points?
Suppose I have set of $n$ multi-variate polynomial equations with $n$ unknowns $x_1, \dots, x_n$. The $n$ equations have real coefficients and are quadratic (so largest degree is $2$). How do I ...
2
votes
0
answers
124
views
Surjectivity of a restricted Hitchin map
Let $C$ be a smooth projective algebraic curve over the complex numbers and let $V$ be a stable rank 2 vector bundle on $C$. Then consider the vector bundle $\mathcal{E}nd_0(V) $ of tracefree ...
6
votes
0
answers
135
views
For a proper scheme $X,$ the neutral component of automorphism group scheme $\mathrm{Aut}^0_X$ is an algebraic group
I am currently reading through M. Brion's "Notes on Automorphism Groups of Projective Varieties" found here. On page 5 he writes
We assume from now on that $X$ is proper, and we denote by $\...
- 101
4
votes
1
answer
289
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Counter example to every closed subscheme $\operatorname{Proj} A$ is of the form $\operatorname{Proj}A/I$
I was under the impression that for a positively graded ring $A$ (not necessarily generated in degree $1$) that every closed subscheme of $\operatorname{Proj}A$ was of the $\operatorname{Proj}A/I$. ...
- 391