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Christopher D. Hacon

Publications and source records attributed to Christopher D. Hacon.

At least 37 records · Page 2Linked to original sources
arXiv2013

On the three dimensional minimal model program in positive characteristic

Let $f:(X,B)\to Z$ be a 3-fold extremal dlt flipping contraction defined over an algebraically closed field of characteristic $p>5$, such that the coefficients of $\{B\}$ are in the standard set $\{1-\frac 1n|n\in \mathbb N\}$, then the flip of $f$ exists. As a consequence, we prove the existence of minimal models for any projective $\Q$-factorial terminal variety $X$ with pseudo-effective canonical divisor $K_X$.

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arXiv2012

Existence of log canonical closures

Let $f:X\to U$ be a projective morphism of normal varieties and $(X,Δ)$ a dlt pair. We prove that if there is an open set $U^0\subset U$, such that $(X,Δ)\times_U U^0$ has a good minimal model over $U^0$ and the images of all the non-klt centers intersect $U^0$, then $(X,Δ)$ has a good minimal model over $U$. As consequences we show the existence of log canonical compactifications for open log canonical pairs, and the fact that the moduli functor of stable schemes satisfies the valuative criterion for properness.

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arXiv2012

On log canonical inversion of adjunction

We prove a result on the inversion of adjunction for log canonical pairs that generalizes Kawakita's result to log canonical centers of arbitrary codimension.

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arXiv2011

Singularities of pluri-theta divisors in Char $p>0$

We show that if $(X,Θ)$ is a PPAV over an algebraically closed field of characteristic $p>0$ and $D\in |mΘ|$, then $(X,\frac 1 m D)$ is a limit of strongly $F$-regular pairs and in particular ${\rm mult}_x(D)\leq m\cdot \dim X$ for any $x\in X$.

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arXiv2011

The Sarkisov program

Any two birational Mori fibre spaces are connected by a sequence of Sarkisov links.

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arXiv2009

Factoring 3-fold flips and divisorial contractions to curves

We show that 3-fold terminal flips and divisorial contractions to a curve may be factored by a sequence of weighted blow-ups, flops, blow-downs to a locally complete intersection curve in a smooth 3-fold or divisorial contractions to a point.

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arXiv2009

Deformations of canonical pairs and Fano varieties

This paper is devoted to the study of various aspects of deformations of log pairs, especially in connection to questions related to the invariance of singularities and log plurigenera. In particular, using recent results from the minimal model program, we obtain an extension theorem for adjoint divisors in the spirit of Siu and Kawamata and more recent works of Hacon and McKernan. Our main motivation however comes from the study of deformations of Fano varieties. Our first application regards the behavior of Mori chamber decompositions in families of Fano varieties: we prove that, in the case of mild singularities, such decomposition is rigid under deformation when the dimension is small. We then turn to analyze deformation properties of toric Fano varieties, and prove that every simplicial toric Fano variety with at most terminal singularities is rigid under deformations (and in particular is not smoothable, if singular).

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arXiv2008

Singularities on normal varieties

In this paper we generalize the definitions of singularities of pairs and multiplier ideal sheaves to pairs on arbitrary normal varieties, without any assumption on the variety being Q-Gorenstein or the pair being log Q-Gorenstein. The main features of the theory extend to this setting in a natural way.

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arXiv2008

On Ueno's Conjecture K

We show that if $X$ is a smooth complex projective variety with Kodaira dimension $0$ then the Kodaira dimension of a general fiber of its Albanese map is at most $h^0(Ω^1 _X)$.

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arXiv2008

On the geography of threefolds of general type

Let $X$ be a complex nonsingular projective 3-fold of general type. We show that there are positive constants $c$, $c'$ and $m_1$ such that $χ(ω_X)\geq -c\Vol (X)$ and $P_m(X)\geq c'm^3\Vol (X)$ for all $m\geq m_1$.

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arXiv2006

On the geography of Gorenstein minimal 3-folds of general type

Let $X$ be a minimal projective Gorenstein 3-fold of general type. We give two applications of an inequality between $χ(ω_X)$ and $p_g(X)$: 1) Assume that the canonical map $Φ_{|K_X|}$ is of fiber type. Let $F$ be a smooth model of a generic irreducible component in the general fiber of $Φ_{|K_X|}$. Then the birational invariants of $F$ are bounded from above. 2) If $X$ is nonsingular, then $c_1^3\leq {1/27} c_1c_2+{10/3}$ where $c_1$, $c_2$ are Chern invariants of $X$.

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arXiv2004

Holomorphic one-forms on varieties of general type

It has been conjectured that varieties of general type do not admit nowhere vanishing holomorphic one-forms. We confirm this conjecture for smooth minimal varieties and for varieties whose Albanese variety is simple.

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