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 DCP2 as a pair (P2, D) of a surface together with a divisor, and allow both the surface and the curve to degenerate. For plane curves of degree d ≥ 4, we obtain a compactification Md 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 - KX is ample and a divisor D in a given linear system on X with specified singularities. Note that X may be non-normal, and KX is π-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 Md is smooth in this case.