A Compactification of the Space of Plane Curves

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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 P² 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 at least 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 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 Md is smooth in this case.