El. knyga: Inflectionary Invariants for Isolated Complete Intersection Curve Singularities

"We investigate the role played by curve singularity germs in the enumeration of inflection points in families of curves acquiring singular members. Let N [ greater than or equal to] 2, and consider an isolated complete intersection curve singularity germf : (CN, 0) [ arrow] (CN[ minus]1, 0). We define a numerical function m [ x in a square] [ arrow] ADm(2)(f) that naturally arises when counting mth-order weight-2 inflection points with ramification sequence (0, . . . , 0, 2) in a 1-parameter family of curves acquiring the singularity f [ equals] 0, and we compute ADm(2)(f) for several interesting families of pairs (f,m). In particular, for a node defined by f : (x, y) [ x in a square] [ arrow] xy, we prove that ADm (2)(xy) [ equals] [ x in a square](m[ plus]1)4 [ x in a square], and we deduce as a corollary that ADm (2)(f) [ greater than or equal to] (mult0 [ delta symbol]f ) [ x in a square](m[ plus]1)4 [ x in a square] for any f, where mult0 [ delta symbol]f is the multiplicity of the discriminant [ delta symbol]f at the origin in the deformation space. Significantly, we prove that the function m [ x in a square] [ arrow] ADm (2)(f)[ minus](mult0 [ delta symbol]f ) [ x in a square](m[ plus]1)4 [ x in a square] is an analytic invariant measuring how much the singularity "counts as" an inflection point. We prove similar results for weight-2 inflection points with ramification sequence (0, . . . , 0, 1, 1) and for weight-1 inflection points, and we apply our results to solve a number of related enumerative problems"--