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On the dimension of the Hilbert scheme of curves. (English) Zbl 1195.14004
Let \(\text{Hilb}^{P}({X})\) be the Hilbert scheme of smooth connected non-degenerate curves \(C \subset X\) with Hilbert polynomial \(P(m)=dm+1-g\) where \(X\) is either \({\mathbb P}^3\), \({\mathbb P}^4\) or a smooth quadric threefold \(Q\) in \({\mathbb P}^4\). Let \(U\) be any irreducible component of \(\text{Hilb}^{P}({X})\).
In this paper, the author gives a good lower bound for the dimension of \(U\) for \(X= {\mathbb P}^3\) in the range \(g^2 \geq d^3\) (Theorem 1.3). In proving this result, the author needs to calculate the Euler characteristic \( \chi(N_{C/S})\) of the normal sheaf where \(S\) is a smooth surface containing \(C\) and the variation of \( \chi(N_{C/S})\) which we need when \(S\) is singular (the latter is not easy). If \(X=Q\) the author shows that if \(g^2 > d^3/2\) (resp. \(g^2 < 4d^3/1125\)) up to lower degree terms, then \(\dim U \) is always greater than the expected value \(3d\) (resp. there exists a component with \(\dim U = 3d\)). To get the existence result he uses smoothing techniques introduced by E. Sernesi [Invent. Math. 75, 25–57 (1984; Zbl 0541.14024)].
Finally recall that a curve \(C \subset {\mathbb P}^r\) is called rigid if every deformation of \(C\) is induced by an automorphism of \({\mathbb P}^r\). For \(X={\mathbb P}^4\) the author shows that there are no rigid curve in a range asymptotically given by \(g^2 > 9d^3\), thus contributing to a conjecture of J. Harris and I. Morrison [Moduli of curves. New York, NY: Springer (1998; Zbl 0913.14005)], stating that only normal rational curves are rigid.
Reviewer’s remark: In Theorem 2.1 there is a missing number 1 in Peskine-Gruson’s result on the maximum genus. This makes Proposition 2.2 slightly inaccurate; one should there replace \(g\) by \(g-1\), or exclude e.g. complete intersections in the conclusion. To correct the inaccuracy, one may in Theorem 1.3 redefine \(\mu\) by replacing \(g\) by \(g-1\) in the definition.

14C05 Parametrization (Chow and Hilbert schemes)
14H10 Families, moduli of curves (algebraic)
14H50 Plane and space curves
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