Given a web (multifoliation) 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 numbers of lines and disjoint lines contained in a projective surface and tangent to a contact distribution.
"Extactic Divisors for Webs and Lines on Projective Surfaces." Michigan Math. J. 67 (4) 743 - 756, November 2018. https://doi.org/10.1307/mmj/1531447376