15
votes
Accepted
Is there a classification of minimal algebraic threefolds?
It depends what you mean by classification.
The key results for surfaces IMO are: 1) Any surface $S$ of general type has a canonical model given by $S_{can}:={\rm Proj} R(K_S)$ and a unique minimal ...
- 2,342
13
votes
On Q-Cartier Divisors
Maybe I can say something useful here. The main confusion seems to be how to find the sheaves/ideals/modules associated to multiples of divisors. As Martin Bright points out, symbolic power of a ...
- 19.7k
10
votes
Accepted
Is being of general type stable under generization
The answer is yes to the original question and is a theorem of Noboru Nakayama in his book "Zariski decomposition and abundance" Theorem VI.4.3, which I state here for convenience:
Theorem (...
- 2,906
10
votes
Accepted
$K_X+B \equiv 0$ implies $K_X + B \sim_\mathbb{Q} 0$?
For lc pair or slc pair, it is true. This is Gongyo’s result. See [J. ALGEBRAIC GEOMETRY 22 (2013) 549–564].
BTW, the relative version is also true, which is not a trivial generalization of the ...
- 1,054
7
votes
Accepted
Generic Smoothness Type of Results in Positive Characteristic
Correction. I just realized that there are examples where the geometric generic fiber is NOT generically reduced. In all of my comments and the answer below, I was assuming that the geometric ...
6
votes
References for the minimal model program
A very light introduction is contained in A first glimpse at the minimal model program. (Also available here.)
I would also second Simone's suggestion: The first two chapters of Kollár-Mori are ...
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6
votes
References for the minimal model program
Of course it depends mostly on your background. But the first chapter, as well as the first half of the second chapter of Kollár-Mori's "Birational Geometry of Algebraic Varieties" is an incredibly ...
- 7,722
6
votes
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$m$-th root of holomorphic section of direct image of relative line bundle
If I understand the question correctly, then here is a likely answer.
But before getting there, let me say that this is a very poorly formed question. If you are asking for help, then put at least as ...
- 41.6k
6
votes
Accepted
Relative logarithmic cotangent bundle
First of all, it's unclear what you mean by $\Omega^1_{X_0}(\log D)$ since $X_0$ is singular.
Second, if you make up such a definition then most probably such a vector bundle will not exist. Note ...
- 14.6k
5
votes
Accepted
How to split a Multi-section into finitely many Sections via base-change?
First, you have a multisection $D_1 = D \times_Y Y'$ for the family $X' \to Y'$, which is still generically finite of the same degree over $Y'$.
On the other hand, let $D' = D \times_{Y''} Y'$. Then ...
- 34k
5
votes
Accepted
Picard number of a general fiber of a fiber contraction
I do not think so, because of the following result.
Proposition. A smooth del Pezzo surface $F$ can be realised as the general fibre of a Mori fibre space if and only if it is not isomorphic to the ...
- 63.7k
5
votes
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Bertini's type theorems over imperfect fields
I needed to know the answer to this myself, so here is a good reference:
Hubert Flenner, Liam O’Carroll, and Wolfgang Vogel, Joins and
intersections, Springer Monographs in Mathematics, Springer-...
- 1,543
5
votes
Is there a purely inseparable covering $\mathbb{A}^2 \to K$ of a Kummer surface $K$ over $\mathbb{F}_{p^2}$?
See Proposition 4.5 of my paper with D. Abramovich, "Lang’s Conjectures, Fibered Powers, and Uniformity", New York J. Math. 2 (1996) 20–34.
A supersingular elliptic curve in characteristic $p>2$ ...
- 29.9k
5
votes
Accepted
Singularities of contractions of extremal faces
In characteristic 0, the answer is well known. By assumption there is an ample divisor $A$ such that $K_X+\Delta+A$ cuts out $F$ and hence by the BPF theorem $K_X+\Delta+A\sim _{\mathbb Q,f}0$ and in ...
- 2,342
4
votes
Accepted
Derived category of singular varieties
Let $\tilde{X}_k$ be the normalization of the closed $k$-codimension stratum, so $\tilde{X}_0$ is the normalization of $X$. Then there is a diagram of pullback functors between the categories $\text{...
- 7,812
4
votes
Accepted
Termination of a minimal model program
We'll show a more general statement. Suppose $(X,\Delta)$ has klt singularities and $f : Y \to X$ is a projective birational morphism with $Y$ normal and $\mathbb{Q}$-factorial. Suppose further that $...
- 2,870
4
votes
Accepted
Existence of terminal $3$-fold flips
Yes - there are very many such examples, and you can cook up examples by a procedure called 'Mori's algorithm'.
A k2A flipping neighbourhood is a 3-fold flipping contraction $f\colon(C\subset X)\to (P\...
- 1,036
3
votes
Flatness of Fano Contractions
I don’t believe this is true; the following example comes from Debarres’s “Higher-dimensional algebraic geometry”. Take $C$ a curve of genus $g$, $d\geq g$, and let $C_d\to J^d(C)$ be the Abel—Jacobi ...
- 200
3
votes
Intuition behind Kawamata's definition of a relative movable Cartier divisor
Let me try to say something that might be useful.
At the risk of stating the obvious, the motivation is to extend the notion of movable divisor (class) to the relative setting. If you haven't already,...
- 100
3
votes
Accepted
Tie-Breaking Trick for Log Canonical Pairs and F-pure pairs in Positive Characteristic
We claim the following holds.
Proposition (cf. [Tan17, Prop. 3 and Idea of Thm. 1; Wan, Proof of Thm. 3.5, Case 2]). Let $X$ be a complete normal variety $X$ of dimension $d$ over an infinite perfect ...
- 1,543
3
votes
Accepted
A Decomposition for Iitaka fibration
If $X$ is smooth (projective over the complex numbers), then $R(K_X)$ is finitely generated by BCHM. We may thus assume that $R(kK_X)$ is generated in degree 1 for some $k>0$. Passing to a log ...
- 2,342
3
votes
Accepted
Controlling singularities on log mmp
If $(X,D)$ is terminal and the stable base locus of $K_X+D$ contains no components of the support of $D$, then any sequence of steps $f:X\to X'$ of the $K_X+D$ MMP yields a terminal pair $(X',D'=f_*D)$...
- 2,342
3
votes
Accepted
relative tangent sheaf
I am not sure I understand the second question, but the answer to the first one is no. Take for $f$ the blowing up of a smooth curve $C$ in $\mathbb{P}^3$. Then $f$ is the projective bundle $\mathbb{P}...
- 35.4k
3
votes
Accepted
Log resolution of a variety of log general type
If $K_{\tilde B}+\tilde \Delta=f^*(K_B+\Delta)+E$ where $E$ is effective and exceptional, then $h^0(m(K_{\tilde B}+\tilde \Delta))=h^0(m(K_B+\Delta))$ for any $m\geq 0$ and hence also the Kodaira ...
- 2,342
3
votes
Accepted
Two morphisms possess the same Viehweg's variation
Since the definition only depends on the general fiber, and $\beta$ is birational, one may assume that $\beta$ is actually and isomorphism.
So, then $L$ is defined as a subfield with minimal ...
- 41.6k
2
votes
rational effective implies effective?
Jun Yan found a counter example for weak del pezzo surface.
Let $X$ be weak del pezzo surface of degree $4$, Let $X=X_{4,4A_1}$, irreducible $(-2)$-curves are
$E_1-E_2,L_{123}=L-E_1-E_2-E_3,E_4-E_5,...
- 1,666
2
votes
Accepted
Small contraction for Hyperkähler Varieties
Let $f:X\to Y$ be a birational contraction where $X$ is hyperkähler, then $K_X\sim 0$ and $K_Y=f_*K_X\sim 0$, and hence $K_X=f^*K_Y$. In particular, this means that $Y$ has canonical singularities. ...
- 1,054
2
votes
Accepted
Intuition behind Kawamata's definition of a relative movable Cartier divisor
The base locus of a divisor $D$ on $X$ is the same as those points where $\mathscr O_X(D)$ is not generated by global sections, which can be identified with the locus where the natural map
$$
\tag{$\...
- 41.6k
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