All Questions
22,548 questions
3
votes
2
answers
271
views
Orbits under the automorphism group of projective space
Let $\mathbb{P}^d_K$ be projective space of dimension $d\geq 1$ over an infinite field $K$. Let $x\in\mathbb{P}^d_K$ with $\dim\overline{\lbrace x\rbrace}=n\leq d-1$.
My question: is the set $\lbrace ...
- 133
3
votes
1
answer
413
views
Twists of elliptic curves
I have a few questions regarding twists of elliptic curves.
In the context of the Shafarevich group, I see people refer to the group of twists of an elliptic curve $E/\mathbb{Q}$ by $H^1(\mathbb{Q}, ...
- 2,907
5
votes
1
answer
265
views
Non-existence of power divided structure on a maximal ideal of truncated polynomial rings (example from Koblitz)
In 3.3 of Berthelot-Ogus's book Notes on Crystalline Cohomology, an example from Koblitz is exhibited without proof.
Let $k$ be a field (or a commutative ring) with characteristic $p>0$, $$A:=k[x_1,...
- 441
0
votes
0
answers
54
views
Functional equations with coupled arguments and additive sructure
Let $G$ be a locally compact abelian group and let $f: G \to \mathbb{R}^+$ be a continuous function satisfying the functional equation
$$f(x + \phi(y)) + f(y + \phi(x)) = 1 + f(x+y)$$
for all $x, y \...
2
votes
1
answer
309
views
Serre functors and global dimensions
Let $k$ be a field.
Let $\mathcal{C}$ be an abelian category (over $k$).
We say that $\mathcal{C}$ has a finite global dimension if there exists integer $n > 0$ such that
$$
\operatorname{Ext}^i(M,...
- 889
1
vote
1
answer
141
views
Why is $2A_0(X)=0$ for a cubic threefold $X$ containing a line, over an arbitrary field $k$
I can't quite follow Proposition $2.1$ of "UNIVERSAL UNRAMIFIED COHOMOLOGY OF CUBIC FOURFOLDS CONTAINING A PLANE". I posted this on Math stackexchange but got no answer.
Let $X$ be a smooth ...
- 679
1
vote
0
answers
92
views
Counting conjugacy classes of completely reducible subgroups in general linear groups [closed]
Let $G = GL(n, q)$ be the general linear group over the finite field $\mathbb{F}_q$ with $q$ elements, where $q$ is a power of a prime $p$. Let $m$ be a positive integer dividing $q-1$. Suppose $\...
7
votes
0
answers
124
views
Projections of closed geodesics under the modular function
In the answers to this question it was shown that for closed geodesics on $\mathbb{H}^2/\Gamma(2)$, the projection under the modular function $\lambda$ is an immersed topological component of a real ...
- 68.9k
5
votes
0
answers
175
views
Is $\overline{\mathcal{M}}_{g,n}$ a Koszul space?
In https://arxiv.org/abs/1902.06318 Dotsenko proved that $\overline{\mathcal{M}}_{0,n+1}$ is a Koszul space, i.e. it is both formal and coformal. Equivalently, a space is Koszul if it is formal and ...
- 641
1
vote
0
answers
84
views
Describing monoidal categories of positive-weight representations geometrically
Let $G=\mathbb{G}_m.$ The monoidal category $\mathcal{C}=\text{Rep}(G)$ of $G$-representations (also known as the category $\text{Gr}$ of graded vector spaces) can be written geometrically as $\...
- 423
1
vote
1
answer
127
views
Vanishing of higher morphisms for pair moduli
Consider a moduli scheme $P$ of sheaves $F$ on a Calabi-Yau n-fold $X$ with a section $s\in H^0(F)$. Alternatively, such objects can be described as maps $\mathcal{O}_X \xrightarrow{s} F$ called pairs....
- 988
6
votes
0
answers
338
views
Deeper meaning behind the occurrence of the factor $\frac{\log q}{i}$ in Deninger's results
In two papers Deninger proved the following:
If $q=p^{n}$ and $p$ is a finite prime of $\mathbb{Z}$, $B=\mathbb{C}[\mathbb{C}]$ is generated by symbols of the form $e^{\alpha}$, $\alpha\in\mathbb{C}$,...
- 1,805
1
vote
1
answer
160
views
Is every log resolution a sequence of blowups?
Suppose we have a variety $X$ over a field of characteristic zero. Choose any ideal sheaf $\mathcal{I}$ on $X$. Is every log resolution of the pair $(X,\mathcal{I})$ a sequence of blow ups? I cannot ...
- 11
0
votes
0
answers
51
views
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 ${\rm dim} V_i =i$....
- 2,974
1
vote
1
answer
82
views
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 $V_p$ ...
- 2,974
3
votes
1
answer
180
views
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
114
views
$\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 ...
- 2,974
1
vote
0
answers
62
views
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
votes
0
answers
120
views
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
272
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
views
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 \...
- 893
8
votes
0
answers
287
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
views
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
views
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
vote
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
votes
0
answers
220
views
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
329
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
views
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
150
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
444
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 ...
- 366
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 $\...
- 6,048
2
votes
0
answers
94
views
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
vote
0
answers
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}...
- 201
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
views
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
322
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
340
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
131
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