Titlebook: Geometry of Algebraic Curves; Volume II with a con Enrico Arbarello,Maurizio Cornalba,Phillip A. Grif Textbook 2011 Springer-Verlag Berlin

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The Hilbert Scheme,and illustrate several pathological behaviors of Hilbert schemes, focussing especially on the case of Hilbert schemes of curves. In particular, we present Mumford’s example of a Hilbert scheme of space curves which is everywhere non-reduced along one of its components.

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Elementary deformation theory and some applications,es, and discuss the local Torelli problem. We close with a couple of loose ends, such as a study of the positivity properties of Hodge bundles and the proof of a theorem of Kempf on deformations of symmetric products of curves.

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Line bundles on moduli,urn to Mumford’s remarkable idea that the Grothendieck Riemann–Roch theorem can be effectively used to produce relations among classes in the moduli spaces of curves. Finally, as a first exemplification of the ideas introduced in the chapter, we determine the Picard group of the closure of the hyperelliptic locus in ..

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,Brill–Noether theory on a moving curve,i’s conjecture, which is the basic result that was announced in Chapter V of Volume I. The second part of the chapter is devoted to the projective realizations of algebraic curves as ramified covers of ?. and as plane curves, and to a number of unirationality results for moduli spaces in low genus.

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