On the existence of certain families of curves (Q794721)

Let \(V\) denote an open irreducible subset of the Hilbert-scheme of \(\mathbb P^r\) parametrizing irreducible and smooth (if \(r\geq 3)\) resp. nodal (if \(r=2)\) curves of genus \(g\) and \(\pi\colon V\to \mathcal M_ g\) the natural map into the moduli space of curves of genus \(g\). \(V\) is called to have the expected number of moduli if \(\dim \pi(V)=\min(3 g-3, 3g - 3+\rho(g,r,n))\), where \(\rho(g,r,n)\) is the Brill-Noether number. It is the aim of the paper to construct such families of curves. For plane curves the result is quite complete. It is shown that for all \(g\) and \(n\geq 5\) such that \(n-2\leq g\leq \binom{n-1}{2}\) there is an irreducible component of the family of plane irreducible nodal curves of degree \(n\) and genus \(g\), having the expected number of moduli. In the range \(n-2\leq g\leq 3n/2-3\) (resp. \(2n-4\leq g\leq \binom{n-1}{2}\) this result was known (resp. independently proved) by \textit{P. Griffiths} and \textit{J. Harris} [Duke Math. J. 47, 233--272 (1980; Zbl 0446.14011)] (resp. by Coppens). For \(r\geq 3\) the result is: for all \(g\) and \(n\geq r+1\) such that \(n-r\leq g\leq(r(n-r)-1)/(r-1)\) (resp. \(n-3\leq g\leq 3n-18\) if \(r=3)\) there is an open set of an irreducible component of the Hilbert scheme of \(\mathbb P^r\) parametrizing smooth irreducible curves of degree \(n\) and genus \(g\), which has the expected number of moduli. An immediate consequence is a special case of a theorem of \textit{D. Eisenbud} and \textit{J. Harris} [Invent. Math. 74, 371--418 (1983; Zbl 0527.14022)] on the existence of very ample line bundles of a prescribed type. The proofs of the theorems are given by induction, the induction step being roughly as follows: Start with a particular curve \(C\) in \(\mathbb P^n\) whose existence is known, construct a new curve by adding a particular rational curve \(\gamma\) and get a new curve \(C'\) by flat smoothing of \(C\cup \gamma\).