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On the Hilbert scheme of smooth curves in \(\mathbb{P}^4\) of degree \(d=g+1\) and genus \(g\) with negative Brill-Noether number. (English) Zbl 1490.14009

Many authors have investigated the claim of F. Severi [Vorlesungen über algebraische Geometrie. (Übersetzung von E. Löffler.). Leipzig-Berlin: B. G. Teubner (1921; JFM 48.0687.01)] that the Hilbert scheme \({\mathcal H}_{d,g,r}\) of smooth connected non-degenerate curves of degree \(d\) and genus \(g\) in \(\mathbb P^r\) is irreducible in the Brill-Noether range \(0 \leq \rho (d,g,r) = g - (r+1)(g-d+r)\), in which case there is a distinguished component of \({\mathcal H}_{d,g,r}\) dominating the moduli space \(\mathcal M_g\) of genus \(g\) curves. While Severi’s claim is false, in each counterexample the locus \({\mathcal H}^L_{d,g,r}\) of linearly normal curves is irreducible and it is possible that this is what Severi intended in the first place (see [C. Ciliberto and E. Sernesi, in: Proceedings of the first college on Riemann surfaces held in Trieste, Italy, November 9-December 18, 1987. Teaneck, NJ: World Scientific Publishing Co.. 428–499 (1989; Zbl 0800.14002)] and Lopez in his Mathematical Review of [C. Keem, Proc. Am. Math. Soc. 122, No. 2, 349–354 (1994; Zbl 0860.14003)]).
In the paper under review the authors study the case \(r=4\) and \(d=g+1\) outside the Brill-Noether range, meaning that \(g \leq 14\). In their previous work [C. Keem and Y.-H. Kim, Arch. Math. 113, No. 4, 373–384 (2019; Zbl 1423.14028)] they showed that \({\mathcal H}_{g+1,g,4}\) is empty for \(g \leq 8\), \({\mathcal H}_{10,9,4}\) is reducible, \({\mathcal H}_{11,10,4} = {\mathcal H}^L_{11,10,4}\) is irreducible, and \({\mathcal H}_{13,12,4}\) is reducible. Therefore they focus here on the cases \(g = 11, 13, 14\), where they show through a case by case analysis that \({\mathcal H}_{g+1,g,4} = {\mathcal H}^L_{g+1,g,4}\) is irreducible and generically reduced.

MSC:

14C05 Parametrization (Chow and Hilbert schemes)
14H10 Families, moduli of curves (algebraic)
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References:

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