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 ...

Community wiki

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 ...

- 41.6k

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

Accepted

### $m$-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

Accepted

### 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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