Details

Edoardo Ballico^[a]

Low genus curves with low degree on a general quintic \(3\)-fold are finite and they have maximal rank

Pages: 271-285
Received: 25 August 2016
Accepted in revised form: 22 May 2017
Mathematics Subject Classification (2010): 14J32, 14M10, 14H50.
Keywords: General quintic \(3\)-fold, curves in Calabi-Yau \(3\)-folds, Clemens' conjecture.
Author address:
[a]: Department of Mathematics, University of Trento, 38123 Povo (TN), Italy

Abstract: Let \(W\subset \mathbb{P}^4\) be a general quintic \(3\)-fold. Fix integers \(d, g\) with \(1\le g \le 3\) and \(g+3\le d \le 11\). In this paper we prove that \(W\) contains only finitely many smooth curves \(C\subset \mathbb{P}^4\) of degree \(d\) and genus \(g\), all of them smooth and isolated points of the Hilbert scheme of \(W\) and that each such \(C\) has maximal rank, i.e. \(h^1(\mathcal{I} _C(t))\cdot h^0(\mathcal{I} _C(t)) =0\) for all \(t\in \mathbb{N}\).

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