Extactic divisors for webs and lines on projective surfaces

TL;DR

This paper introduces extactic divisors for webs and lines on projective surfaces, providing new tools to analyze contact loci and reinterpreting classical results about lines on surfaces.

Contribution

It constructs divisors associated with webs and linear systems to study abnormal contact, and applies these to classical and new problems involving lines on surfaces.

Findings

Reobtained Salmon's classical result on lines on surfaces

Analyzed the number and configuration of lines tangent to contact distributions

Developed a new approach using extactic divisors for webs

Abstract

Given a web (multi-foliation) and a linear system on a projective surface we construct divisors cutting out the locus where some element of the linear system has abnormal contact with the leaf of the web. We apply these ideas to reobtain a classical result by Salmon on the number of lines on a projective surface. In a different vein, we investigate the number of lines and of disjoint lines contained in a projective surface and tangent to a contact distribution.