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15 votes
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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 ...
Hacon's user avatar
  • 2,382
11 votes
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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 (...
Frank's user avatar
  • 2,956
10 votes
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$K_X+B \equiv 0$ 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 ...
Chen Jiang's user avatar
  • 1,084
8 votes

Application of MMP in other branches of algebraic geometry

Birational experts on this website will probably give you a more detailed answer, but I think that the "techniques" developped to solve the MMP, rather than the big results, are more likely ...
Libli's user avatar
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6 votes
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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 ...
Piotr Achinger's user avatar
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 ...
Sándor Kovács's user avatar
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 ...
diverietti's user avatar
  • 7,862
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 ...
Sándor Kovács's user avatar
6 votes
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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 ...
Sasha's user avatar
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6 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-...
Takumi Murayama's user avatar
6 votes

About the contractability

Artin's contractibility criterion [1] implies that a necessary and sufficient condition to contract an irreducible curve $E$ on a surface $X$ is that $E^2 <0 $. In general, the contraction will ...
Francesco Polizzi's user avatar
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 $p>2$ ...
Felipe Voloch's user avatar
5 votes
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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 ...
Francesco Polizzi's user avatar
5 votes
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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\...
Tom Ducat's user avatar
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5 votes
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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 ...
Hacon's user avatar
  • 2,382
4 votes
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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{...
Dmitry Vaintrob's user avatar
4 votes
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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 $...
Dori Bejleri's user avatar
  • 2,900
4 votes

Two different resolution of a three fold

The strict transform of the plane $x = z = 0$ in one of the resolutions is still a plane, and in the other it is isomorphic to the blowup of the plane.
Sasha's user avatar
  • 37.6k
3 votes
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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}...
abx's user avatar
  • 37.4k
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 ...
Devlin Mallory's user avatar
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,...
Bort's user avatar
  • 100
3 votes
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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 ...
Takumi Murayama's user avatar
3 votes
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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 ...
Hacon's user avatar
  • 2,382
3 votes
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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 ...
Hacon's user avatar
  • 2,382
3 votes
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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 ...
Sándor Kovács's user avatar
3 votes

Mori cones and projective morphisms

The argument is correct, and in fact can be easily extended to the case where $f$ is generically finite. Indeed, we have $$f_*([X]) = (\deg f) [Y],$$ so the projection formula [Fulton, Prop. 8.3(c)] ...
R. van Dobben de Bruyn's user avatar
3 votes
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Mori cones and projective morphisms

It suffices to assume that $f$ is surjective (equivalently, dominant). Then for $C \subset Y$ any irreducible curve there exists an irreducible curve $D \subset X$ such that $f(D) = C$. (A schemy ...
2 votes
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Central fibre singularities

It could easily happen that $X_0$ has log terminal singularities and $X$ is not log terminal. The standard example is if $f:Y\to X$ is a flipping contraction of a 3-fold over a curve $T$ (where the ...
Hacon's user avatar
  • 2,382

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