Sphere Packings, Lattices and Groups

These are notes from a jointly taught class at the University of Chicago and lectures by the first author in Santa Cruz. Topics covered include: construction of moduli spaces of stable maps, Gromov-Witten invariants, quantum cohomology, and examples. These notes will appear in the proceedings of the 1995 Santa Cruz conference.

Notes on stable maps and quantum cohomology

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A mathematical theory of quantum cohomology

We introduce a method of constructing the virtual cycle of any scheme associated with a tangent-obstruction complex. We apply this method to constructing the virtual moduli cycle of the moduli of stable maps from n-pointed genus g curves to any smooth projective variety. As an application, we give an algebraic definition of GW-invariants for any smooth projective variety.

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Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties

Riemann surfaces: Basic definitions Functions and maps More examples of Riemann surfaces Integration on Riemann surfaces Divisors and meromorphic functions Algebraic curves and the Riemann-Roch theorem Applications of Riemann-Roch Abel's theorem Sheaves and Cech cohomology Algebraic sheaves Invertible sheaves, line bundles, and $\check H^1$ References Index of notation Index of terminology.

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Algebraic Curves and Riemann Surfaces

On extremal rational elliptic surfaces

Triple Covers in Algebraic Geometry

In this article we exhibit certain projective degenerations of smoothK3 surfaces of degree 2g−2 in ℙ
 g
 (whose Picard group is generated by the hyperplane class), to a union of two rational normal scrolls, and also to a union of planes. As a consequence we prove that the general hyperplane section of suchK3 surfaces has a corank one Gaussian map, ifg=11 org≥13. We also prove that the general such hyperplane section lies on a uniqueK3 surface, up to projectivities. Finally we present a new approach to the classification of prime Fano threefolds of index one, which does not rely on the existence of a line.

Projective degenerations of K3 surfaces, Gaussian maps, and Fano threefolds

In this article we address the problem of computing the dimenlsion of the space of plane curves of degree d with n general points of multiplicity m. A conjecture of Harbourne and Hirschowitz implies that when d > 3m, the dimension is equal to the expected dimension given by the Riemann-Roch Theorem. Also, systems for which the dimension is larger than expected should have a fixed part containing a multiple (-1)-curve. We reformulate this conjecture by explicitly listing those systems which have unexpected dimension. Then we use a degeneration technique developed to show that the conjecture holds for all m < 12.

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Rick Miranda

Papers

Algebraic Curves and Riemann Surfaces

On extremal rational elliptic surfaces

Triple Covers in Algebraic Geometry

Projective degenerations of K3 surfaces, Gaussian maps, and Fano threefolds

Linear systems of plane curves with base points of equal multiplicity