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2
votes
2answers
209 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 ...
13
votes
1answer
535 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 ...
5
votes
1answer
406 views

Nagata's conjecture in positive characteristic

For a $\mathbb C P^2$ is known a result: if through the generic points $p_1,p_2,\dots p_n$ with multiplicities $m_1,m_2\dots, m_n$ correspondingly a degree $d$ irreducible reduced curve passes then ...
5
votes
3answers
487 views

Does every linear group admit a subgroup of dimension 1?

Suppose that $G$ is a linear group of positive dimension, defined over some field $k$. Is that true, that $G$ admits a (closed) one-dimensional subgroup? I'm pretty much sure this is true in ...
1
vote
1answer
133 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( ...
4
votes
1answer
314 views

Properties of divisors when moving from char 0 to char p.

Consider a smooth projective variety $X$ over $\mathbb{C}$ such that $X$ has models over $\mathbb{Z}[1/N]$ and $X_p=X_{\mathbb{Z}[1/N]}\times \text{Spec}(\mathbb{F}_p)$ is also a smooth projective ...
6
votes
1answer
294 views

Vanishing theorems in positive characteristic

In the paper Deligne, Pierre; Illusie, Luc (1987), "Relèvements modulo $p^{2}$ et décomposition du complexe de De Rham", Inventiones Mathematicae 89 (2): 247–270, doi:10.1007/BF01389078 I found the ...
2
votes
0answers
181 views

Lang isogeny for group stacks

Let $G$ be a commutative algebraic group stack over $\mathbb{F}_q$ (I don't really care about the precise definition: I'm secretly thinking about the Picard stack of a projective curve). To what ...
2
votes
0answers
231 views

Kawamata-Viehweg Vanishing Theorem for Excellent Surfaces

Is there any version of Kawamata-Viehweg vanishing theorem that holds for excellent surfaces? I will be very glad if there is one such, but if not then is it at least true for a surface over a non ...
3
votes
0answers
241 views

Adjunction Formula for Weil Divisors on a Normal Variety X

Let $X$ be a normal variety over an algebraically closed field $k$ of characteristic $p>0$ and $S$ be a prime Weil divisor on $X$ which is normal too. Now if $K_X+S$ is NOT $\mathbb{Q}$-Cartier, ...
0
votes
0answers
175 views

How to Construct a ''Nice'' Birational Model in Characteristic p>0

Let $X=Spec\ A$ be a normal affine variety of dimension $n$ over an algebraically closed field $k$ of characteristic $p>0$. Also, assume that $S\subset X$ be a prime Weil-divisor on $X$. Now, I ...
2
votes
1answer
162 views

On a Strongly F-regular Pair (X, \Delta)

Let $X$ be a normal projective variety over a field of characteristic $p>0$ and $(X, \Delta\geq 0)$ be a pair such that $K_X+\Delta$ is $\mathbb{Q}$-Cartier whose index is not divisible by $p$. ...
7
votes
1answer
291 views

Commutativity of the Chow ring in positive characteristic

I was looking in Ravi Vakil's notes on Intersection Theory, Class 20, where he introduces the bivariant intersection theory, in particular the Chow ring $A^\ast (X)$. On p. 2, he writes the following ...
1
vote
0answers
148 views

“Higher” Tangent spaces in char-p geometry - definition?

Hi, everyone! I have some construction that requires exact definition. I considered polynomial homomorphisms from $\Bbbk$ to $U_n(\Bbbk)$ (unitriangular matrices), where $char \Bbbk = p>0$ (more ...
2
votes
1answer
446 views

Heisenberg group in characteristic two

I have seen two constructions called the Heisenberg group. If $k$ is a field of characteristic not equal to $2$ and $V$ is a $2n$-dimensional vector space over $k$ with symplectic form $\omega : V ...
2
votes
2answers
301 views

Is the absolute Galois group of the field of Laurent series in positive characteristic finitely generated?

If $K$ is an algebraically closed field of characteristic $p>0$, then $K((t))$, the field of Laurent series with coefficients in $K$, has infinitely many Galois extensions of degree $p$. Indeed, ...
1
vote
0answers
126 views

Classification of fibres in pencils of curves of genus two

For the case of characteristic positive, there exist some clasification of families of curves of genus two over a elliptic curve?.
2
votes
0answers
303 views

understanding Milne's article “Duality in the flat cohomology of a surface”

I have trouble understanding a point of Milne's article "Duality in the flat cohomology of a surface" http://jmilne.org/math/articles/1976a.pdf see the "Alternatively" on p. 177, paragraph before ...