Non-reduced components of the Hilbert scheme of curves using triple covers (Q6566712)

Let \(H_{d,g} (\mathbb P^r)\) denote the Hilbert scheme parametrizing curves \(C \subset \mathbb P^r\) of degree \(d\) and genus \(g\) and let \(\mathcal I_{d,g,r} \subset H_{d,g} (\mathbb P^r)\) corespond to the union of irreducible components whose general member is a smooth, irreducible, non-degenerate curve. \textit{D. Mumford} used families of curves on cubic surfaces to produce a generically non-reduced component in \(\mathcal I_{14,24,3}\), the first such example [Am. J. Math. 84, 642--648 (1962; Zbl 0114.13106)]. \textit{J. O. Kleppe} generalized the idea, producing \(s\)-maximal families of space curves with \(s=3\) [Lect. Notes Math. 1266, 187--207 (1987; Zbl 0631.14022)] and \textit{J. O. Kleppe} and \textit{J. C. Ottem} made more examples with \(s=4\) and \(s=5\) [Int. J. Math. 26, No. 2, Article ID 1550017, 30 p. (2015; Zbl 1323.14005)]. \textit{A. Dan} used Hodge loci and flag Hilbert schemes to produce more examples on surfaces of arbitrary degree \(s\) [Math. Nachr. 290, No. 17--18, 2800--2814 (2017; Zbl 1387.14027)]. An example with \(r > 3\) was constructed by Ciliberto, Lopez and Miranda [\textit{C. Ciliberto} et al., in: Higher dimensional complex varieties. Proceedings of the international conference, Trento, Italy, June 15--24, 1994. Berlin: Walter de Gruyter. 167--182 (1996; Zbl 0893.14007)].\N\NHere the authors make more non-reduced components. The main theorem says that if \(e,\gamma \in \mathbb Z\) with \(\gamma \geq 3\) and \(e \geq 4 \gamma + 5\), then for \(g = 3 \gamma + 3 e, d = 3 e + 1, r = e-\gamma+1\), the Hilbert scheme \(\mathcal I_{d,g,r}\) has a component \(\mathcal H\) of dimension \(r^2+7e+4\) such that at a general point \([X] \in \mathcal H\), \(\dim T_{[X]} \mathcal H = \dim \mathcal H + 1\), hence \(\mathcal H\) is generically non-redcued. The general member \(X\) of the family corresponding to \(\mathcal H\) lies on a cone \(F\) over a smooth curve \(Y\) of degree \(e\) and genus \(\gamma\) in \(\mathbb P^{r-1}\) such that \(X\) is projectively normal and passes through the vertex \(P\) of \(F\), there is a ruling of \(F\) that is tangent to \(X\) at \(P\) with multiplicity two, and projection from \(P\) to the hyperplane containing \(Y\) induces a triple cover \(X \to Y\). The proof uses a characterization of smooth curves \(X\) on the cone \(F\) passing through the vertex and a description of a free resolution for \(\mathcal I_X\) given by \textit{M. V. Catalisano} and \textit{A. Gimigliano} [J. Pure Appl. Algebra 135, No. 3, 225--236 (1999; Zbl 0937.14015)] and arguments similar to those of \textit{C. Ciliberto} [Math. Z. 194, 351--363 (1987; Zbl 0595.14004)] and Ciliberto, Lopez and Miranda [loc. cit.] to show that deformations of \(X\) give curves in the same family.