Genus bound of curves on surfaces of almost minimal degree

Wanseok Lee; Euisung Park

doi:10.1016/j.jpaa.2025.107891

Genus bound of curves on surfaces of almost minimal degree

Wanseok Lee
, Euisung Park

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let C⊂P^r, r≥3, be a nondegenerate projective integral curve of degree d and arithmetic genus g. Castelnuovo theory says that (i) if g>π₁(d,r) then C is contained in a surface of minimal degree, and (ii) if g>π₂(d,r) then C is contained in a surface of degree ≤r. In this paper, we prove that if g>π₀(d,r+1)+1 then C is contained in a surface of minimal degree or a del Pezzo surface. To this aim, we show that π₀(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 π₀(d,r+1)+1.

Original language	English
Article number	107891
Journal	Journal of Pure and Applied Algebra
Volume	229
Issue number	2
DOIs	https://doi.org/10.1016/j.jpaa.2025.107891
Publication status	Published - 2025 Feb

Bibliographical note

Publisher Copyright:
© 2025 Elsevier B.V.

Keywords

Castelnuovo theory
Del Pezzo surface
Surface of almost minimal degree
Surface of minimal degree

ASJC Scopus subject areas

Algebra and Number Theory

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10.1016/j.jpaa.2025.107891

Cite this

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title = "Genus bound of curves on surfaces of almost minimal degree",

abstract = "Let C⊂Pr, r≥3, be a nondegenerate projective integral curve of degree d and arithmetic genus g. Castelnuovo theory says that (i) if g>π1(d,r) then C is contained in a surface of minimal degree, and (ii) if g>π2(d,r) then C is contained in a surface of degree ≤r. In this paper, we prove that if g>π0(d,r+1)+1 then C is contained in a surface of minimal degree or a del Pezzo surface. To this aim, we show that π0(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 π0(d,r+1)+1.",

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author = "Wanseok Lee and Euisung Park",

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doi = "10.1016/j.jpaa.2025.107891",

language = "English",

volume = "229",

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T1 - Genus bound of curves on surfaces of almost minimal degree

AU - Lee, Wanseok

AU - Park, Euisung

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N2 - Let C⊂Pr, r≥3, be a nondegenerate projective integral curve of degree d and arithmetic genus g. Castelnuovo theory says that (i) if g>π1(d,r) then C is contained in a surface of minimal degree, and (ii) if g>π2(d,r) then C is contained in a surface of degree ≤r. In this paper, we prove that if g>π0(d,r+1)+1 then C is contained in a surface of minimal degree or a del Pezzo surface. To this aim, we show that π0(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 π0(d,r+1)+1.

AB - Let C⊂Pr, r≥3, be a nondegenerate projective integral curve of degree d and arithmetic genus g. Castelnuovo theory says that (i) if g>π1(d,r) then C is contained in a surface of minimal degree, and (ii) if g>π2(d,r) then C is contained in a surface of degree ≤r. In this paper, we prove that if g>π0(d,r+1)+1 then C is contained in a surface of minimal degree or a del Pezzo surface. To this aim, we show that π0(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 π0(d,r+1)+1.

KW - Castelnuovo theory

KW - Del Pezzo surface

KW - Surface of almost minimal degree

KW - Surface of minimal degree

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DO - 10.1016/j.jpaa.2025.107891

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JO - Journal of Pure and Applied Algebra

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ER -

Genus bound of curves on surfaces of almost minimal degree

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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