Seminar Algebra and Geometry: Enrico Schlesinger (Politecnico di Milano)

On the problem of connectedness for the Hilbert schemes of space curves

Grothendieck has constructed the Hilbert scheme  that parametrizes all subschemes of P^N with a given Hilbert polynomial, and Hartshorne in his thesis has shown that it is connected. For a  curve in P³ the Hilbert polynomial is given by the degree d and by the arithmetic genus g. I will explain why, even if one is primarily interested in smooth curves, the correct class of curve to look at is that of locally Cohen Macaulay curves. These are parametrized by an open subscheme H_d,g of the full Hilbert scheme.

It is an open question whether H_d,g is connected whenever nonempty. This question was motivated by a result by Martin-Deschamps and Perrin: they showed that H_d,g always has an irreducible component made up by "extremal curves"; these curves have the largest cohomology among curves in H_d,g. So there is no obstruction from semicontinuity that prevents the possibility that any smooth curve be specialized to an extremal curve.

I will discuss the state of affairs about this question, and briefly describe work in progress (with the help of Macaulay 2) showing that curves of type (a,a+4) on a smooth quadric surface are in the connected component of extremal curves; this problem was  raised in Hartshorne's papers   "On the connectedness of the Hilbert scheme of curves in P³" Comm. Alg. 28, 2000  and "Questions of connectedness of the Hilbert scheme of curves in P³" in the volume for Abhyankar's 70th (2004), and was still on the open problems list of the Workshop "Components of the Hilbert Schemes" (AIM Palo Alto 2010).

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