Algebraic geometry studies properties of specific algebraic varieties, on the one hand, and moduli spaces of all varieties of fixed topological type on the other hand. Of special importance is the moduli space of curves, whose properties are subject of ongoing research. The rationality versus general type question of these and related spaces is of classical and also very modern interest with recent progress presented in the conference. Certain different birational models of the moduli space of curves and maps have an interpretation as moduli spaces of singular curves and maps. For specific varieties a wide range of questions was addressed, including extrinsic questions (syzygies, the k-secant lemma) and intrinsic ones (generalization of notions of positivity of line bundles, closure operations on ideals and sheaves).
Cite this article
David Eisenbud, Frank-Olaf Schreyer, Ravi Vakil, Claire Voisin, Classical Algebraic Geometry. Oberwolfach Rep. 7 (2010), no. 2, pp. 1573–1623DOI 10.4171/OWR/2010/27