Hacon
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$\chi(\omega_X)>0$ implies that $X$ is of general type
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14 votes

If it is not of general type, then the Iitaka fibration $X\to Z$ is fibered by tori say $F$ (the fibers of the Iitaka fibration have Kodaira dimension 0 and maximal Albanese dimension, and so by a ...

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Is canonical model always with canonical singularity
13 votes

I believe that $(Y,B)$ is always klt for some boundary $B$. In fact by Theorem 5.2 of https://projecteuclid.org/download/pdf_1/euclid.jdg/1090347529, after passing to a truncation of the ...

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Understanding what it means to be ''of general type''
9 votes

$X$ is of general type iff it is birational to its canonical model $X^c={\rm Proj}(\oplus _{m\geq 0}H^0(mK_X))$. Here $X^c$ has canonical singularities and $mK_{X^c}$ is a very ample Cartier divisor ...

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Vector bundles on abelian varieties
9 votes

How about the following example in Debarre's paper "ON COVERINGS OF SIMPLE ABELIAN VARIETIES" available on his web page. On page 5, it says: "...if L is an ample line bundle on an abelian variety A of ...

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A paradox on the deformation of singularities
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8 votes

I don't think it is true that $\mathcal X$ is $\mathbb Q$-Gorenstein. Suppose in fact that $\dim \mathcal X _t=2$ for all $t\in C$ and $\mathcal X \to Z$ is a flipping contraction with exceptional ...

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Can an abelian variety dominate a variety of general type?
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8 votes

Theorem: Let $A$ be an abelian variety and $f:A\to X$ a dominant morphism to a projective variety of general type, then $\dim X=0$. Proof: Replacing $f$ by a birational model of the Stein ...

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Applications of the boundedness of birational automorphisms
8 votes

I don't have any direct/immediate consequence of the main result of the paper. However the techniques that we develop here are very useful. In work in progress, we plan to use these techniques to ...

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Is there a precise relationship between the goals of moduli theory and the minimal model program?
6 votes

Koll\'ar's paper "Moduli of varieties of general type" https://arxiv.org/abs/1008.0621 provides an excellent introduction. We now know that moduli spaces of canonical models of varieties of general ...

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Sequences of divisors satisfying Serre vanishing?
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6 votes

Claim: $D_*$ is SV iff for any ample divisor $H$ there exists $m(H)$ such that if $m\geq m(H)$, then $D_m-H$ is nef. Suppose $D_*$ is SV. Fix $A$ very ample. Since $D_*$ is SV, then $H^i(D_m-H-jA)=0$ ...

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Counterexamples to Kollár's conjecture
6 votes

I just saw this question (better late than never). The following references come to mind (the first one has an example which somewhat improves on the example of Ein-Lazarsfeld, the others have a ...

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Vanishing theorem for big divisors
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6 votes

Let $X\to \mathbb P ^1$ be a family of smooth quadric surfaces degenerating to $X_0=\mathbb F _2=\mathbb P (\mathcal O _{\mathbb P ^1}\oplus \mathcal O _{\mathbb P ^1} (2))$ where $0\in \mathbb P ^1$. ...

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Numerically equivalent effective divisors and semiampleness
5 votes

Does Example 2.4 from http://www.math.utah.edu/~hacon/commA.dvi answer your question?

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Rational contraction and Proj of section ring
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4 votes

Let's assume that $|kD|=|kM|+kF$ where $|M|$ is base point free and moreover $Sym ^kH^0(M)\to H^0(kM)$ is surjective for any $k>0$. This can be achieved replacing $X$ by an appropriate resolution ...

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Curves on a Kahler manifold
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4 votes

From "Bounded sets of sheaves on Kähler manifolds" By Matei Toma, J. reine angew. Math. 710 (2016), 77–93 Lemma 4.4. Let X be a Kähler manifold, r be an integer and F be a set of compact reduced ...

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Bertini theorem for big divisors and klt pairs
4 votes

If $X$ is $\mathbb P ^2$ blown up at a point $O$ with exceptional divisor $E$ and $D=H+E$ where $H$ is the pull-back of the hyperplane class, then any effective divisor $D'\equiv D$ will vanish along $...

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A question about running MMP with scaling
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4 votes

We have $K_X+\Delta +\lambda C=f^*(K_Z+\Delta '+\lambda C')$ by the Base Point Free Thm (3.3 and 3.7(4) in Koll\'ar-Mori 1998). Clearly $K_Z+\Delta '+\lambda C'$ is nef. If $f^+:X^+\to Z $ is the flip,...

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Log resolution of a variety of log general type
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3 votes

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 ...

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Log canonical counterexample to Kawamata-Viehweg vanishing
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3 votes

In Prop. 3.13 of http://arxiv.org/pdf/1212.5105.pdf there is an example of a generically finite map $\lambda : T\to A$ where $A$ is an abelian variety and $T$ is a Gorenstein variety with log ...

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A Decomposition for Iitaka fibration
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3 votes

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 ...

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Controlling singularities on log mmp
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3 votes

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 $(X',D'=f_*D)$...

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Multiplication maps for big line bundles
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2 votes

If $R(L)=\oplus H^0(mL)$ is not finitely generated, the above surjectivity will fail, however it will hold "asymptotically" for any big line bundle $L$. In fact, by Fujita's approximation of ...

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What are the most important instances of the "yoga of generic points"?
2 votes

There is a spectacular proof of the ACC for log canonical thresholds in $\mathbb C ^N$ due to Ein, de Fernex and Mustata https://arxiv.org/abs/0905.3775 that relies on taking generic limits of ...

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Fundamental group of Log del Pezzo surfaces
2 votes

I would proceed as follows. Replacing $S$ by its minimal resolution I have a weak log Fano pair $(S,B)$ which is klt and $-(K_S+B)$ is semiample and big. In particular $-K_S\sim _Q -(K_S+B)+B$ is big ...

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Decompose a big divisor as nef big divisor and effective divisor
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2 votes

I think the answer is (almost, i.e. up to birational modification) yes but non-trivial see Theorem 1.3 of arXiv:1208.4150 "ACC for log canonical thresholds" by Christopher Hacon, James McKernan, ...

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Non-unique completion of a flat family of smooth projective varieties
1 votes

I believe that this never happens. The reason is as follows. The morphism $f:X\to Y$ is a small birational morphism (it contracts some divisors on the special fiber) and so by standard arguments $Y$ ...

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Polarization of an abelian variety made by the sum of two divisors
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1 votes

For any $P\in Pic^0(A)$ consider the map $|\Theta +P|\times |D-P|\to |\Theta +D|\cong \mathbb P ^2$. Since $D$ is effective, there is an abelian subvariety $T\subset Pic ^0(A)$ such that $|D-P'|\ne \...

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Finiteness of Gorenstein indexes and volumes for varieties in a bounded family
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1 votes

By Noetherian induction, it suffices to show that indicies and volumes are bounded over an open subset of any irreducible component of $T$. We may assume that $T$ is smooth and there is a dense set $\...

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Central fibre singularities
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1 votes

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 ...

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Dimension of para-pluricanonical systems
1 votes

Actually, I think the above argument always works because the minimal model has mild singularities (or one can reason directly on $X$; see ref below). Let $X$ be a smooth variety of general type and $...

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A key step in Del Busto's effective Matsusaka theorem
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1 votes

It seems to me that $[\frac 1nF_n]$ is exc. and there is a short exact sequence inducing $$W(k)→V(k+1)→H^1((−∑k_iE_i)|_{\sum \eta _iE_i})$$ which has positive degree on a bunch of (non-reduced $\...

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