Content summary:

• 14.10: Yoneda's lemma, representable functors, the functor of points of projective space;

• 21.10: fiber products, representable arrows and their properties; sheaves; projective bundles;

• 28.10: Grassmannians: representability, tangent bundle, Plücker embedding; 27 lines on a smooth cubic;

• 04.11: the functor of points of the Hilbert scheme; flatness, numerical invariants, cohomology and base-change;

• 11.11: Castelnuovo-Mumford regularity (from sheaf cohomology and syzygies) and its properties;

• 18.11: Mumford's universal polynomials, existence of flattening stratifications, Hilb and Quot functors: two representability statements;

• 25.11 (1h): revision on the Hilbert scheme and basic examples (hypersurfaces, lines, conics, twisted cubics);

• 02.12: construction of the Quot scheme, the quasi-projective case, spaces of morphisms;

• 09.12: overview of infinitesimal deformation theory: functors of Artin rings, prorepresentable functors, tangent-obstruction theories;

• 16.12: cancelled;

• 13.01: computation of a tangent-obstruction theory for the Quot scheme, statement of Schlessinger's theorem;

• 20.01: tangent-obstruction for Hilb; (pointwise) definition of the Hilbert-Chow morphism; the case of curves; the case of C^2: relation to quiver representations and hyperKähler reduction; incidence correspondences and connectedness; Fogarty's theorem (Hilb is a resolution of singularities of Sym(surface)); the Hilbert scheme of four points on a threefold is singular; statement of Göttsche's formula and the Heisenberg algebra action;

• 27.01: configurations of ordered points on P^1 - the open locus and Knudsen's construction of a compactification; the universal curve and gluing morphisms; why automorphisms prevent representability by a scheme (in higher genus);

• 03.02: (locally trivial) deformations of (smooth) varieties, nodal curves, and smoothness of \bar M_{0,n}.

Essential bibliography:

• construction of the Quot scheme: Nitsure's notes, Sernesi's book;

• infinitesimal deformation theory: Fantechi and Göttsche's notes in FGA explained, Sernesi's book, Talpo and Vistoli's notes;

• Hilbert schemes of points: Fantechi and Göttsche's notes, Nakajima's book;

• \bar M_{0,n}: Cavalieri's notes, Talpo and Vistoli's notes.

Prerequisites: a course in algebraic geometry at the level of Hartshorne ch. II and III

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Lecture notes