The study of the birational geometry of various moduli spaces of curves with the help of the computer algebra system Macaulay

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The moduli space of curves Mg is the universal parameter space for algebraic curves (Riemann surfaces) of given genus, in the sense that its points correspond to isomorphism classes of curves of genus g. The study of the geometry and topology of Mg is a central problem in algebraic geometry. A well-known principle, due to Mumford, asserts that all moduli spaces parameterizing curves of genus g ≥ 2 (with or without marked points or level structures), are varieties of general type, with a finite number of exceptions that occur in relatively small genus, when these varieties tend to be uniruled, or even unirational. A variety X is said to be uniruled, when through a general point x Є X there passes a rational curve ƒ : P1 ( X, whereas X is said to be of general type, when the canonical bundle Kx has the maximum number of sections. From the point of view of classification theory, uniruledness is the opposite of being of general type. The aim of this project is to compute the Kodaira dimension of various covers covers of Mg. These include universal Picard varieties and moduli spaces classifying pairs consisting of a curve together with a point of order l in the Jacobian variety respectively. The proofs are expected to rely on syzygy calculations assisted by the Macaulay system.