Canonical singularities of orders over surfaces
We classify the possible ramification data and etale local structure of orders over surfaces with canonical singularities.
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arXiv subjects
Paul Hacking
Publications and source records attributed to Paul Hacking.
Lectures on flips and minimal models
This document contains notes from the lectures of Corti, Kollár, Lazarsfeld, and Mustaţă at the workshop ``Minimal and canonical models in algebraic geometry" at MSRI, Berkeley, April 2007. The lectures give an overview of the recent advances on canonical and minimal models of algebraic varieties obtained by Hacon--McKernan and Birkar--Cascini--Hacon--McKernan.
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Degenerations of del Pezzo surfaces I
Let X be a surface with quotient singularities which admits a smoothing to the plane. We prove that X is a deformation of a weighted projective plane P(a^2,b^2,c^2), where a,b,c is a solution of the Markov equation a^2+b^2+c^2=3abc. We also prove a generalisation for del Pezzo surfaces of degree K^2 at least 5.
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Compactification of the moduli space of hyperplane arrangements
Consider the moduli space M^0 of arrangements of n hyperplanes in general position in projective (r-1)-space. When r=2 the space has a compactification given by the moduli space of stable curves of genus 0 with n marked points. In higher dimensions, the analogue of the moduli space of stable curves is the moduli space of stable pairs: pairs (S,B) consisting of a variety S (possibly reducible) and a divisor B=B_1+..+B_n, satisfying various additional assumptions. We identify the closure of M^0 in the moduli space of stable pairs as Kapranov's Chow quotient compactification of M^0, and give an explicit description of the pairs at the boundary. We also construct additional irreducible components of the moduli space of stable pairs.
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Compact moduli of hyperplane arrangements
The minimal model program suggests a compactification of the moduli space of hyperplane arrangements which is a moduli space of stable pairs. Here, a stable pair consists of a scheme X which is a degeneration of projective space and a divisor D=D_1+..+D_n on X which is a limit of hyperplane arrangements. For example, in the 1-dimensional case, the stable pairs are stable curves of genus 0 with n marked points. Kapranov has defined an alternative compactification using his Chow quotient construction, which may be described fairly explicitly. We prove that these two compactifications coincide. We deduce a description of all stable pairs.
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Semistable divisorial contractions
The semistable minimal model program is a special case of the minimal model program concerning 3-folds fibred over a curve and birational morphisms preserving this structure. We classify semistable divisorial contractions which contract the exceptional divisor to a normal point of a fibre. Our results can be applied to describe compact moduli spaces of surfaces.
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Compact moduli of plane curves
We construct a compactification M_d of the moduli space of plane curves of degree d. We regard a plane curve C as a surface-divisor pair (P^2,C) and define M_d as a moduli space of pairs (X,D) where X is a degeneration of the plane. We show that, if d is not divisible by 3, the stack M_d is smooth and the degenerate surfaces X can be described explicitly.
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A Compactification of the Space of Plane Curves
We define a geometrically meaningful compactification of the moduli space of smooth plane curves, which can be calculated explicitly. The basic idea is to regard a plane curve D in P^2 as a pair (P^2,D) of a surface together with a divisor, and allow both the surface and the curve to degenerate. For plane curves of degree d at least 4, we obtain a compactification M_d which is a moduli space of stable pairs (X,D) using the log minimal model program. A stable pair (X,D) consists of a surface X such that -K_X is ample and a divisor D in a given linear system on X with specified singularities. Note that X may be non-normal, and K_X is Q-Cartier but not Cartier in general. We give a rough classification of stable pairs of arbitrary degree, a complete classification in degrees 4 and 5, and a partial classification in degree 6. The compactification is particularly simple if d is not a multiple of 3 - in particular the surface X has at most 2 components. We give a characterisation of these surfaces in terms of the singularities and the Picard numbers of the components. Moreover, we show that M_d is smooth in this case.
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