# Tag Info

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 ...
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### 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 ...
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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 (...
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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 ...
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### 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 ...

### 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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### 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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-...
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### 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 $p>2$ ...
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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 ...
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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 ...
• 2,342
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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 ...
• 41.6k