A NOTE ON THE HILBERT SCHEME OF CURVES OF DEGREE d AND GENUS

This note is inspired by a lecture given during the school “Liason theory and related topics” and contains a summary of the results in [15] about the connectedness of the Hilbert scheme of curves of degree d and genus (d−3 2 ) − 1. The only novelty is the list of degrees for which smooth and irreducible curves appear. This short note was inspired by a talk I gave at the Politecnico of Torino during the School “Liaison theory and related topics”. The question of the connectedness of the Hilbert schemes Hd,g of locally Cohen–Macaulay curves C ⊂ P 3 of degree d and arithmetic genus g arose naturally after Hartshorne proved in his PhD thesis that the Hilbert scheme of all one dimensional schemes with fixed Hilbert polynomial is connected. The result is somewhat too general since, even to connect one smooth curve to another, it involves curves with embedded or isolated points. On the other hand, if the question is addressed under the more restrictive hypothesis of smooth curves, then the Hilbert scheme need not be connected: a counterexample can be found for (d, g) = (9, 10). In the recent years, after the developing of liaison theory, it has become clear that, even though one can be interested in the classification of smooth curves, the natural class to look at is the class of locally Cohen–Macaulay curves, i.e. the class of schemes of equidimension 1 with all their local rings Cohen–Macaulay. In other words, they are 1 dimensional schemes with no embedded or isolated points. The answer to the question in case of locally Cohen– Macaulay curves is known, so far, only for low degrees or high genera. The scheme Hd,g is non empty when d ≥ 1 and g = (d−1 2 ) (that corresponds to the case of plane curves), or d > 1 and g ≤ (d−2 2 ) . After the paper [9], it is well known that Hd,g contains an irreducible component consisting of extremal curves (i.e. curves having the largest possible Rao function). This is the only component for d ≥ 5 and (d − 3)(d − 4)/2+ 1 < g ≤ (d − 2)(d − 3)/2 while in the cases d ≥ 5, g = (d − 3)(d − 4)/2 + 1 and d ≥ 4, g = (d − 3)(d − 4)/2 the Hilbert scheme is not irreducible, but it is connected (see [1], [12]). The connectedness is trivial for d ≤ 2 since the scheme is irreducible, see [5], while it has been proved for d = 3, d = 4 and any genus in [11], [13] respectively. Note that for d = 3, 4 there is a large number of irreducible components: they are approximatively 1 3 |g| for d = 3 and 1 24 g 2 for d = 4. The paper [4] has given a new light to the problem, in fact Hartshorne provides some methods to connect particular classes of curves to the irreducible component of extremal curves, while in the paper [14] Perrin has proved that all the curves whose Rao module is Koszul can be connected to the components of extremal curves. This note deals with the first unknown case for high genus, i.e. g̃ = (d − 3)(d − 4)/2 − 1 and its purpose is to give an overview of the results in the forthcoming [15]. Since it contains only a brief state of the art, for a more complete treatment of the topic the reader is referred to [4], [5]. In [15] we have studied the connectedness of the Hilbert scheme Hd,g̃ of locally Cohen– Macaulay curves in P3 = Pk , where k is an algebraically closed field of characteristic zero. A way one can follow to prove the connectedness of Hd,g̃, is to first identify its irreducible com-