On the existence of components of the Hilbert scheme with the expected number of moduli
Let W be an open irreducible subset of the Hilbert scheme of \({\mathbb{P}}^ r\) parametrizing smooth irreducible curves of degree n and genus g and \(\pi\) : \(W\to {\mathcal M}_ g\) the natural map to the moduli space of curves of genus g. W is said to have the expected number of moduli if \(\dim(\pi (W))=\min \{3g-3,3g-3+\rho \}\), where \(\rho =\rho (g,n,r)\) is the Brill-Noether number. The purpose of the paper is the construction of such families of curves for negative \(\rho\) 's, extending the previous existence range shown by \textit{E. Sernesi} [Invent. Math. 75, 25-57 (1984; Zbl 0541.14024)]. For \(r\geq 12\) or \(r=10\), it is shown the existence of families having the expected number of moduli in the range \(-(5r+\epsilon)g/(4r+9- \epsilon)+f(r)\leq \rho \leq 0\) where \(\epsilon\) is defined by \((r- \epsilon)/3=\lceil (r-1)/3\rceil\) and f(r) is a rational function of r asymptotically like \(5/4r^ 2\) (for the explicit expression see section 1 of the paper). - Note that the range where the existence problem makes sense is \(\rho \geq -3g+3\) and that the above result gives an affirmative answer roughly for \(\rho \geq -5(g-r^ 2)/4.\) The proof has two parts: (a) A careful study of the normal bundle of general nonspecial curves in \({\mathbb{P}}^ r\). This is accomplished by studying the special case of nonspecial curves lying in suitable rational normal surface scrolls; (b) an inductive proof starting from curves \(C\subset {\mathbb{P}}^ r\) whose existence was proved by E. Sernesi [loc. cit.] and attaching general nonspecial curves \(\Gamma\) at \(r+4\) general points. The results in (a) are then used to show that this construction is possible and that the curves \(C\cup \Gamma\) are flatly smoothable (the latter by standard deformation theory techniques developed by E. Sernesi [loc. cit.]).
scientific article; zbMATH DE number 4150379