Abstract

A Hilbert scheme is a general concept for describing spaces of subvarieties of a given algebraic variety. As such many familiar spaces in algebraic geometry, such as Grassmann manifolds, can be realized as Hilbert schemes. Hilbert schemes are useful tools in moduli problems of stable curves and their enumerative properties in algebraic geometry with conjectural applications to Gromov-Witten and Donaldson-Thomas invariants. In recent years Hilbert schemes of points have been intensively investigated and these efforts have led to important advances including the solution of some major problems in combinatorics related to symmetric functions. The purpose of this series of lectures is to give a self-contained introduction to Hilbert schemes. The theory will be demonstrated by means of concrete applications. While no prior expertise in algebraic geometry is required, some familiarity with schemes and sheaves will be helpful.



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