Enrico Arbarello

Geometry of algebraic curves

Preface.- Guide to the Reader.- Chapter IX. The Hilbert Scheme.- Chapter X. Nodal curves.- Chapter XI. Elementary deformation theory and some applications.- Chapter XII. The moduli space of stable curves.- Chapter XIII. Line bundles on moduli.- Chapter XIV. The projectivity of the moduli space of stable curves.- Chapter XV. The Teichmuller point of view.- Chapter XVI. Smooth Galois covers of moduli spaces.- Chapter XVII. Cycles on the moduli spaces of stable curves.- Chapter XVIII. Cellular decomposition of moduli spaces.- Chapter XIX. First consequences of the cellular decomposition .- Chapter XX. Intersection theory of tautological classes.- Chapter XXI. Brill-Noether theory on a moving curve.- Bibliography.- Index.

Calculating cohomology groups of moduli spaces of curves via algebraic geometry

We compute the first, second, third, and fifth rational cohomology groups of the moduli space of stable n-pointed genus g curves, for all g and n, using (mostly) algebro-geometric techniques.

Petri's approach to the study of the ideal associated to a special divisor

The ideal defining a canonical curve was classically studied by Max Nofither, Enriques, Petri and Babbage [B]. Recently the subject was again brought to light by Bernard Saint-Donat and David Mumford. The fundamental result is that the ideal of a canonical, non-hyperelliptic, curve C, of genus p >3, is generated by quadratic forms, with the exception of two cases: when C lies on a non-singular ruled surface of degree p 2 in IP p1 (in which case C is trigonal), or else when p = 6 and C is contained in the Veronese surface of IP s (in which case C has a g2). More generally it is natural to ask what can be said about the ideal I . D of an irreducible curve C ~ IP ~1 of genus p, on which the hyperplanes cut a complete linear series IDI. When D is non-special and deg D>3p+ 1 the answer is simple: I . D is generated by quadrat ic forms (see [M-2]). When D is special the situation is more complicated. To this case is devoted the central part of this paper, which closely follows Petri s approach. The result is the following. Take a basis x 1 . . . . . x z of H~ and a basis Yl . . . . . Yt, of H~ The xiy]s can be naturally viewed as elements of H~ spanning a sub-vector space of H~ whose codimension we call r. Choos ing z, . . . . . z~H~ so that the x~y]s and the z~s generate H~ we consider relations of the following type:

Mukai's program for curves on a K3 surface

Let C be a general element in the locus of curves in M_g lying on some K3 surface, where g is congruent to 3 mod 4 and greater than or equal to 15. Following Mukais ideas, we show how to reconstruct the K3 surface as a Fourier-Mukai transform of a Brill-Noether locus of rank two vector bundles on C.

Explicit Brill–Noether–Petri general curves

Let

Teichmüller space via Kuranishi families

p_1,\dots, p_9

Enumerative Geometry of Curves

be the points in

On hyperplane sections of K3 surfaces

\mathbb A^2(\mathbb Q)\subset \mathbb P^2(\mathbb Q)

The Hilbert Scheme

with coordinates