All Questions
13 questions
0
votes
0
answers
89
views
Contraction of extremal ray on a smooth projective threefold
I have some issues about understanding the contraction of extremal ray in a concrete situation:
Let $\mathcal{E}=\mathcal{O}_{\mathbb{P}^1\times \mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1\times \...
- 41
2
votes
0
answers
96
views
Reference request The support of $f$-nef divisor
I'm seaching for a proof of the theorem below.
Do you know any reference?
Consider $f:X\rightarrow Y$ projective birational map between normal varieties and $\mathbb{R}$ cartier divisor $D$ whose ...
- 328
3
votes
0
answers
605
views
Minimal model vs canonical model of a surface
When I have a projective surface $X$, for simplicity smooth, I can find a simpler smooth surface on its binational class. In this way we find in a finite number of steps the simplest surface $Y$, i.e. ...
0
votes
1
answer
325
views
On the birational equivalent class of algebraic surfaces with Picard number $1$
An open subset $U$ of a projective surface $Z$ is big if $\mathrm{codim}_Z(Z\setminus U)\geq2$.
Let $X$ and $Y$ be smooth complex projective surface. If there exists a birational map $f:X\...
- 828
3
votes
0
answers
135
views
Is the generalized Kummer threefold rational in characteristics 3?
Let $E_i\!: y_i^2 = x_i^3 - x_i$, $i = 1, 2, 3$ be three copies of the supersingular elliptic curve in characteristics $3$. Consider on $E_i$ the following automorphism of order $3$:
$$
\sigma(x_i,...
- 1,833
7
votes
1
answer
424
views
Is there a purely inseparable covering $\mathbb{A}^2 \to K$ of a Kummer surface $K$ over $\mathbb{F}_{p^2}$?
Let $E_i\!: y_i^2 = x_i^3 + a_4x_i + a_6$ be two copies ($i = 1$, $2$) of a supersingular elliptic curve over a finite field $\mathbb{F}_{p^2}$, for odd prime $p > 3$. Consider the Kummer surface $...
- 1,833
1
vote
0
answers
262
views
Section ring $R(X,D)$ of $D$ is finitely generated if $\kappa (X,D) \leq 1$
In the remark on the bottom of page 5 of this paper, the author states that
It is well-known fact that the algebra of a Divisor $D$ with $\kappa (X,D) \leq 1$ is finitely generated over $k$. In ...
- 61
5
votes
0
answers
243
views
Map associated to linear system onto curve is morphism
In Mumford's first paper on Surfaces in char $p$ [1], part 2 Step (II), he wants to show that, given an indecomposable curve of canonical type $D$ on a smooth projective surface $F$ with $p_g(F)=0, ...
- 61
2
votes
1
answer
715
views
Castelnuovo and Artin contractibility criteria for families
In the course of a proof, I need some version of Castelnuovo and Artin's contractibility criteria for a family of surfaces. Say I have a family (flat is probably needed, in order to compare ...
- 625
8
votes
1
answer
318
views
rational effective implies effective?
Let $X$ be a weak del pezzo surface, I Wonder whether the following statment is true:
Let $L$ be a line bundle on $X$, then $h^0(L)=0$ implies $h^0(nL)=0$ for all $n\geq 1$.
- 1,982
5
votes
1
answer
482
views
Number of minimal models of a surface
I would like to know if the following statement is true or false:
Given a non-singular complex projective surface $S$, it has at most a countable number of minimal models (up to isomorphism).
We ...
- 1,237
5
votes
1
answer
1k
views
Minimal Model Program for surfaces over algebraically closed fields of characteristic p
Let $k$ be an algebraically closed field of characteristic $p>0$.
I have been trying to find out unsuccessfully if there is a mmp for algebraic surfaces over $k$. I know minimal surfaces are ...
- 2,215
7
votes
3
answers
970
views
Basepoints in the canonical system of algebraic surfaces
Let $X$ be a smooth projective variety defined over $\mathbb{C}$. In the context of the minimal model program it is often important to understand the geometry of the maps defined by the complete ...
- 2,097