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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.

##### MSC:
 14C05 Parametrization (Chow and Hilbert schemes) 14H10 Families, moduli of curves (algebraic) 14H50 Plane and space curves
##### Keywords:
Hilbert scheme; deformation; space curve; rigid curve; smoothing
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