Reducibility of the Hilbert scheme of smooth curves and families of double covers
Families of curves in projective space have long been in studied in algebraic geometry. In the modern language, let \(H_{d,g} (\mathbb P^r)\) denote the Hilbert scheme parametrizing curves \(C \subset \mathbb P^r\) of degree \(d\) and arithmetic genus \(g\) and let \({\mathcal I}_{d,g,r} \subset H_{d,g} (\mathbb P^r)\) be the union of irreducible components whose general member is a smooth irreducible non-degenerate curve. Almost one hundred years ago \textit{F. Severi} claimed that \({\mathcal I}_{d,g,r}\) is irreducible if \(d \geq g+r\) [Vorlesungen über algebraische Geometrie. Leipzig u. Berlin: B. G. Teubner (1921; JFM 48.0687.01)]. While \textit{L. Ein} showed that the claim is true for \(r=3\) [Ann. Scient. Ec. Norm. Sup. 19, 469--478 (1986; Zbl 0606.14003)] and \(r=4\) [\textit{L. Ein}, Proc. Symp. Pure Math. 46, 83--87 (1987; Zbl 0647.14012)], [\textit{E. Mezzetti} and \textit{G. Sacchiero}, Lect. Notes Math. 1389, 183--194 (1989; Zbl 0716.14012)] and [\textit{C. Keem}, Proc. Am. Math. Soc. 122, No. 2, 349--354 (1994; Zbl 0860.14003)] produced counterexamples for \(r \geq 6\): these examples have been produced by exhibiting families of curves whose general member is an \(m\)-fold cover of \(\mathbb P^1\) for \(m \geq 3\). In the paper under review, the authors produce new irreducible components of \({\mathcal I}_{d,g,r}\) whose general member \(C \subset \mathbb P^r\) has the property that the morphism defined by the linear system \(\mathbb P H^0(C, \omega_C (-1))\) has degree two (possibly after removing base points).
Hilbert scheme of smooth connected curves
scientific article; zbMATH DE number 6871333
scientific article