All Questions
624 questions
3
votes
0
answers
287
views
Does the invariant from resolution of singularities provide a Whitney stratification?
The topic of Whitney stratifications came up in a lecture, and the general procedure in the examples was to decompose the singular locus of the variety into the strata starting with the "worst" ones. ...
- 251
5
votes
1
answer
1k
views
Excellent schemes
I noticed that many results in positive characteristic assumes that the object of the theorem is excellent. I have looked up the definition of excellent and have tried to get a feeling for it, but all ...
- 6,700
34
votes
2
answers
3k
views
The work of E. Artin and F. K. Schmidt on (what are now called) the Weil conjectures.
I was reading Dieudonne's "On the history of the Weil conjectures" and found two things that surprised me. Dieudonne makes some assertions about the work of Artin and Schmidt which are no doubt ...
- 50.7k
7
votes
1
answer
762
views
Bertini's Theorem
Let $p_1,...,p_n\in\mathbb{P}^{N}$ be general points. Consider the linear system $|L|$ of hypersurfaces of degree $d$ in $\mathbb{P}^{N}$ with prescribed multiplicities $m_1,...,m_n$ at $p_1,...,p_n$. ...
- 66.3k
2
votes
1
answer
2k
views
Log Canonical pairs
Let $X$ be a normal scheme ad $D = \sum_id_iD_i\subset X$ be a $\mathbb{Q}$-divisor such thay $K_X+D$ is $\mathbb{Q}$-Cartier. Let $f:Y\rightarrow X$ be a log resolution of the pair $(X,D)$ and let us ...
- 20.5k
11
votes
2
answers
918
views
On a proposition in Hartshorne's paper "Ample vector bundles on curves"
In Prop. 4.1, p. 87 of the article "Ample vector bundles on curves" (Nagoya Math. J. 43 [1971], 73--89), R. Hartshorne states the following:
Let $A$ be an abelian variety [over an alg. closed field $...
- 1,285
3
votes
0
answers
843
views
Projective tangent cones, ordinary singularities and blow-ups
Let $X\subset\mathbb{P}^n$ be a projective variety and let $Y\subset X$ be the singular locus of $X$. Assume that $Y$ is smooth. I would like to know if the following are equivalent:
$X$ has an ...
CommunityBot
- 1
24
votes
2
answers
2k
views
Have we ever proved any non-solvable case of reciprocity without the Langlands program ?
The reciprocity of the title is the following not completely well-posed problem:
Fix $P(X)$ a monic irreducible polynomial of degree $n$, with coefficients in $\mathbb Z$. "Describe"
(in some sense) ...
CommunityBot
- 1
0
votes
0
answers
209
views
Resolution of singularities of projective varieties
Let $X\subset\mathbb{P}^n$ be an irreducible variety, and let $Sing(X)$ be its singular locus. Let $Y$ be the blow-up of $\mathbb{P}^n$ along $Sing(X)$. Assume that we know that the strict transform ...
CommunityBot
- 1
13
votes
2
answers
768
views
Is there a proof of Warning's Second Theorem using p-adic cohomology?
Let $\mathbb{F}_q$ be a finite field, $n \in \mathbb{Z}^+$, and $f_1,\ldots,f_r \in \mathbb{F}_q[t_1,\ldots,t_n]$ with $\operatorname{deg}(f_i) = d_i$. Put $d = \sum_{i=1}^n d_i$ and suppose $d< n$...
- 12.9k
2
votes
1
answer
221
views
Resolving nodes of a quintic CY 3-fold
Let's consider the following quintic 3-fold $X$:
\begin{equation}
\{(x_i) \in \mathbb{P}^4 \ | \ x_1f(x)-x_2g(x)=0\}
\end{equation}
for generic homogeneous polynomials $f(x),g(x)$ of degree four. It ...
CommunityBot
- 1
7
votes
0
answers
294
views
Picard scheme of varieties over imperfect fields
Let $k$ be a field and $X$ a proper $k$-scheme. It is a theorem of Murre and Oort that the Picard functor is representable by a $k$-group scheme $\operatorname{Pic}_{X/k}$ which is locally of finite ...
- 4,450
9
votes
1
answer
863
views
Are there noncommutative extensions of $\alpha_p$ by $\mathbb{G}_m$?
Let $k$ be a field of characteristic $p > 0$ (algebraically closed, if you want; that doesn't make a difference). Consider a finite type $k$-group scheme $E$ that is a (central) extension of $\...
- 17.9k
8
votes
1
answer
747
views
Deligne's exterior power
In "Catégories Tannakiennes", Deligne defines the $n$th exterior power of an object $A$ of an abelian tensor category $\mathcal{C}$ as the image of the morphism
$$p : A^{\otimes n} \to A^{\otimes n}, ...
CommunityBot
- 1
6
votes
1
answer
211
views
Recursions for some binary theta series in characteristic 3
Define $A(0), A(1), A(2) \dots$ in ${\bf Z}/3[[x]]$ as follows. For $n$ in $\bf N$ let $s=3^{2n+1}$. Then $A(n) = \sum a_kx^k$ where $a_k$ is the mod 3 reduction of the number of representations of $k$...
- 79.9k
8
votes
3
answers
918
views
Contracting a rational curve in a Calabi-Yau threefold
Let $X$ be a Calabi-Yau threefold and $C \subset X$ be a rational curve with $N_{C/X}\cong \mathcal{O}\oplus \mathcal{O}(-2)$. Can one contract the curve $C$? Assuming the answer is yes, what kind of ...
CommunityBot
- 1
3
votes
0
answers
211
views
Singularities in mixed characteristic
Let $R$ be a regular local ring in mixed characteristic. Moreover, I assume that $R$ is the local ring of a point on a smooth $\mathbb Z_p$-scheme and that $R/pR$ is regular. ($\mathbb Z_p$ is the ...
- 1,590
4
votes
0
answers
209
views
Partial differential Equation over characteristic p
I want some references on partial differential equations over characteristic $p$. If we have a first order partial differential equation, how can we check whether there exists polynomials or rational ...
1
vote
1
answer
687
views
A question on klt pairs
Let $D$ be a $\mathbb{Q}$-divisor in a smooth variety $X$. In Lazarsfeld book "Positivity in Algebraic Geometry 2" I found Proposition 9.5.13 saying that if for any $x\in D$ we have $mult_xD < 1$ ...
- 8,998
10
votes
1
answer
625
views
Can a division algebra have degree divisible by its characteristic?
I apologize in advance if this is easy, but I've tried Googling, and had no luck.
I'm currently working on a proof, and I realized in the course of writing that this proof will break if out there ...
- 44.7k
3
votes
1
answer
391
views
Embedded resolution of curves on smooth varieties
As far as I understand, embedded resolution of singularities means the following: given a variety $X$ over an algebraically closed field, and a closed subvariety $Y$, there exists a birational map $f:...
- 20.5k
4
votes
2
answers
254
views
Invariant planes of a nilpotent matrix with two Jordan blocks of size two
Describe all the invariant 2-dimensional subspaces of $\mathbb{C}^4$ (or $\mathbb{R}^4$) of the nilpotent map
$$
N = \begin{pmatrix}
0 & 1 & & \\
0 & 0 & & \\
& & 0 &...
- 10.5k
2
votes
1
answer
717
views
Singularities of secant varieties of rational normal curves
Let $C\subset\mathbb{P}^n$ be a rational normal curve of degree $n$, and let $Sec_k(C)\subset\mathbb{P}^n$ be its $k$-th secant variety. By Theorem 1.1 in this paper:
http://ac.els-cdn.com/...
CommunityBot
- 1
3
votes
1
answer
276
views
Mumford-Ramanujam examples in characteristic p [and in Arakelov geometry]
For a compact Riemann surface $B$ of genus $\geq 2$, it is a consequence of the Narasimhan-Seshadri theorem that there exist rank-$2$ vector bundles $E \to B$ of degree zero, all of whose symmetric ...
- 13.8k
0
votes
2
answers
490
views
Small birational maps and singularities of the pair
Let $f:X\dashrightarrow Y$ be a small birational map, where $X,Y$ are normal $\mathbb{Q}$-factorial varieties. Let $\Delta_X\subset X$ be an effective $\mathbb{Q}$-divisor such that the pair $(X,\...
CommunityBot
- 1
2
votes
2
answers
480
views
Lifting to char 0, references and questions
Suppose that I have a surface $S$, smooth proper over an algebraically closed (perfect?)field $k$ that lifts algebraically to some $S_W$ defined over a field of char 0. I am interested in properties ...
- 4,111
2
votes
1
answer
202
views
Some Kind of Resolution of Singularites
Let $X \subseteq \mathbb{P}^n$ be a projective variety. I would like to have a morphism $f: \tilde{X}\to X \subseteq \mathbb{P}^n$ where $f$ is finite and birational, $f^* \mathcal{O}_{\mathbb{P}^n}(1)...
CommunityBot
- 1
15
votes
1
answer
1k
views
Number of curves over a finite field
Let $K$ be a finite field. Is there a formula for the number of isomorphism classes of genus $g$ smooth curves over $K$?
In other words does there exists a formula for the number of rational points ...
- 40.3k
2
votes
2
answers
1k
views
Finite Quotients and Resolutions of Singularities
So, I feel like I'm missing something obvious, but I have the following situation:
Let $X\to Y$ be a finite group quotient of schemes (in fact, varieties) by the finite group $G$. Let $\tilde{Y}\to ...
- 5,186
1
vote
1
answer
1k
views
Singular irreducible quadrics
Let $Q\subset\mathbb{P}^n$ be the quadric hypersurface defined by
$$x_0^2+x_1^2+...+x_k^2 =0.$$
If $2\leq k\leq n-1$ then $Q$ is irreducible and $Sing(Q)$ is a linear space of dimension $n-k-1$.
If $...
- 8,998
5
votes
1
answer
704
views
Simultaneous resolution of singularities in special cases of flat families of projective varieties
Let $\pi:\mathcal{X} \to B$ be a flat family of projective varieties. Assume that $B$ is irreducible. Suppose that $\mathcal{X}$ is smooth except for a closed subscheme, say $Y$ which is isomorphic to ...
- 4,111
1
vote
1
answer
156
views
Pushforward of $K_X+D$ on the non-snc locus
Let $f:Y\rightarrow X$ be a birational morphism of smooth projective varieties, $F$ an effective divisor on $X$, $D=f^{-1}F_{\mathrm{red}}+\mathrm{Ex}(f)$, $B$ a smooth subvariety of $Y$ contained in ...
- 4,111
4
votes
2
answers
1k
views
Varieties with big anti-canonical divisor
I recently heard about the following problem:
Let $X$ be a projective variety with klt singularities and such that $-K_X$ is big. Is $X$ a Mori Dream Space ?
Now, $-K_X$ big if and only if $-K_X -\...
- 8,998
7
votes
1
answer
341
views
How can one show that orbit closures in representations of a linear quiver don't have small resolutions?
Let $1\to \cdots\to n$ be a linear quiver of length $n$. Let $\mathbf{d}=(d_1,\dots,d_n)$ be a dimension vector. It's well known (for example, by Gabriel's theorem, but also by basic linear algebra) ...
8
votes
2
answers
741
views
Resolution of unpleasant singularity
I've been working on some varieties defined by taking some quotients of group actions, and the resolutions have been straightforward... until now.
E.g., consider $\mathbb{C}^2$ with the action $(x,y)\...
- 66.3k
4
votes
2
answers
1k
views
Crepant resolution of isolated fourfold singularity
I stumbled upon this isolated singularity of a Calabi-Yau fourfold:
\begin{equation}
x_1x_2+x_3x_4+x_5^2=0
\end{equation}
as a hypersurface in $\mathbb{C}^5$.
Clearly, I can resolve this by a simple ...
CommunityBot
- 1
2
votes
1
answer
386
views
Surfaces singular along a curve
Let $C\subset\mathbb{P}^3$ be a smooth curve a degree $d$ and genus $g$. Let $\mathcal{S}$ be the system of surfaces of degree $k$ in $\mathbb{P}^3$ containing $C$ with multiplicity $\beta$.
What is ...
- 38k
3
votes
2
answers
513
views
Which isolated surface singularity comes from a -5 curve?
Define the surface $X$ to be the total space of $\mathcal{O}_{\mathbb{P}^1}(-5)$.
By contracting the exceptional curve in $X$, we get a surface with an isolated singularity. I am looking for the ...
- 8,998
18
votes
3
answers
3k
views
Lifting varieties to characteristic zero.
If you want to compute crystalline cohomology of a smooth proper variety $X$ over a perfect field $k$ of characteristic $p$, the first thing you might want to try is to lift $X$ to the Witt ring $W_k$ ...
- 8,998
0
votes
1
answer
700
views
Kawamata-Log-Terminal pairs
Let $p_1,...,p_n\in\mathbb{P}^3$ be general points, and let $\Delta\subset\mathbb{P}^3$ be a general surface of degree $d$ with points of multiplicity $m_i$ at $p_i$ for $i = 1,...,n$.
Consider the ...
CommunityBot
- 1
11
votes
1
answer
675
views
Extended Deformation Theory (dg-Lie algebra principle in positive characteristic?)
Recently, I looked at articles that make use of Deligne's idea that "in characteristic 0 every deformation problem is governed by a differential graded
Lie algebra" as explained first in Goldman-...
- 40.3k
3
votes
1
answer
957
views
Is it possible to resolve singularities using only normal varieties?
In characteristic 0, is it possible to have a resolution of singularities where the algebraic varieties at every step of the desingularization process are normal. To be more precise, I would like a ...
- 251
7
votes
0
answers
236
views
Invariant theory of $SL_2$ over a field of positive characteristic
Let $k$ be an algebraically closed field of characteristic $p>0$. Let $W$ be a finite dimensional $SL_2$-module over $k$. Let $V$ be the natural representation of $SL_2$.
What can be said - in ...
- 605
2
votes
1
answer
110
views
Determining the desingularization from the complete local ring
Suppose I have a curve $C$ over a field $k$ and that $p$ is a singular point of $C$. Let $f : X \to C$ be the desingularization of $C$ at $p$. Then for each $s \in f^{-1}(p)$ we have a map of local ...
- 38k
4
votes
1
answer
399
views
Blowing up rational singularities
Let $X$ be a projective surface embedded into $\mathbb{P}^n_{\mathbb{C}}$ having at most rational singularities. Let $\tilde{X} \to X$ be the minimal resolution of $X$. Is it possible to embed $\tilde{...
- 38k
19
votes
3
answers
2k
views
Elkies' supersingularity theorem in higher dimension
The following is a theorem of Elkies:
Let $X$ be an elliptic curve over $\mathbb{Q}$. Then there are infinitely many primes $p$ such that the action of Frobenius on $H^1(\mathcal{O}, X)$ is zero.
...
- 3,597
3
votes
0
answers
287
views
big and small resolutions of singularities of a 4-fold
Suppose we have a projective 4fold hypersurface $X\subset P^n$
with ordinary singularities along a smooth curve $C$, and suppose that there exist a projective small resolution $s:Y\to X$. let us ...
- 3,779
5
votes
0
answers
287
views
Nori fundamental group and etale fundamental group in positive characteristic
Let $X$ be a smooth projective surface over an algebraically closed field of char $p > 0$. Suppose that $\pi_{1}^{et}(X) = \{1\}$. Can Nori fundamental group scheme of $X$ be non-trivial?
- 163
11
votes
0
answers
576
views
What's known about the mod 2 reduction of the level l Jacobi modular equation?
Motivation:
Let $\ell$ be an odd prime. Let $A$ in ${\mathbb Z}/2[[x]]$ be $x+x^9+x^{25}+x^{49}+...$, and $B=A(x^\ell)$. One can use the level $\ell$ Jacobi modular equation to get a polynomial ...
- 5,422
8
votes
1
answer
331
views
If C is a fusion category over a field of nonzero characteristic and dim C = 0, is Z(C) ever fusion?
If $C$ is a fusion category and $\dim(C) \neq 0$ (the latter is automatic in characteristic zero, but not in nonzero characteristic), then the Drinfel'd center $Z(C)$ is fusion. More generally, if $C$...
- 28.1k