On the computation of singular plane curves and quartic surfaces

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 $\mathbb C\mathbb P^3$ having the following combinations of rational double points: $\mathsf D_5\mathsf E_7\mathsf E_7,$ $\mathsf D_7\mathsf D_6\mathsf D_6,$ $\mathsf E_6\mathsf D_8\mathsf D_5,$ $\mathsf E_6\mathsf D_{13},$ $\mathsf E_6\mathsf E_6\mathsf D_7,$ $\mathsf E_6\mathsf E_8\mathsf D_5,$ $\mathsf E_7\mathsf D_6\mathsf D_6,$ $\mathsf E_7\mathsf D_{12},$ $\mathsf E_7\mathsf E_6\mathsf D_6.$ These are all possible cases with total Milnor number 19 which have no point of type $\mathsf A_n.$