Genus bound of curves on surfaces of almost minimal degree

초록

Let C subset of Pr, r >= 3, be a nondegenerate projective integral curve of degree d and arithmetic genus g. Castelnuovo theory says that (i) if g > pi(1)(d, r) then Cis contained in a surface of minimal degree, and (ii) if g > pi(2)(d, r) then Cis contained in a surface of degree <= r. In this paper, we prove that if g > pi(0)(d, r + 1) + 1 then Cis contained in a surface of minimal degree or a del Pezzo surface. To this aim, we show that pi(0)(d, r + 1) + 1 is the upper bound of g when C lies on a surface of degree r which is not a del Pezzo surface. We also provide a specific construction of curves with genus equal to the upper bound pi(0)(d, r + 1) + 1. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

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