Classification of Exceptional Complements: Elliptic Curve Case

TL;DR

This paper classifies certain elliptic curve cases of log del Pezzo surfaces with specific complement properties, advancing understanding in algebraic geometry.

Contribution

It provides a classification of log del Pezzo surfaces with elliptic curve components under specific complement conditions, a novel focus in the field.

Findings

Classification of such surfaces achieved

Identification of elliptic curve components with specific coefficients

Clarification of complement absence in these cases

Abstract

We classify the log del Pezzo surface (S,B) of rank 1 with no 1-,2-,3-,4-, or 6-complements with the additional condition that B has one irreducible component C which is an elliptic curve, and that C has the coefficient b in B with (1/n)floor((n+1)b)=1 for n=1,2,3,4, and 6.