On the computation of singular plane curves and quartic surfaces

2009/06/18 by Carlos Rito, Rito, Carlos · 2 citations
Computer Science · Engineering · Mathematics · #14H50 #14Q05 #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.0906.3480

openalex publication_date 2009/06/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Two Magma functions are given: one computes linear systems of plane curves with non-ordinary singularities and the other computes a scheme which parametrizes given degree plane curves with given singularities. These functions provide an efficient tool to construct explicit equations of singular plane algebraic curves. By computing singular branch curves, we obtain equations of normal quartic surfaces in \mathbb C\mathbb P³ having the following combinations of rational double points: \mathsf D₅\mathsf E₇\mathsf E₇, \mathsf D₇\mathsf D₆\mathsf D₆, \mathsf E₆\mathsf D₈\mathsf D₅, \mathsf E₆\mathsf D₁₃, \mathsf E₆\mathsf E₆\mathsf D₇, \mathsf E₆\mathsf E₈\mathsf D₅, \mathsf E₇\mathsf D₆\mathsf D₆, \mathsf E₇\mathsf D₁₂, \mathsf E₇\mathsf E₆\mathsf D₆. These are all possible cases with total Milnor number 19 which have no point of type \mathsf A_n.

Cited by

• New surfaces with K²=7 and p_g=q≤ 2
• Involutions on surfaces with p_g=q=0 and K²=3

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