Inflectionary Invariants for Isolated Complete Intersection Curve Singularities

Abstract

We investigate the role played by curve singularity germs in the enumeration of inflection points in families of curves acquiring singular members. Let N ≥ 2, and consider an isolated complete intersection curve singularity germ f : (C N , 0) → (C N−1 , 0). We introduce a numerical function m → AD m(2) (f) that arises as an error term when counting m th -order weight-2 inflection points with ramification sequence (0, . . . , 0, 2) in a 1parameter family of curves acquiring the singularity f = 0, and we compute AD m(2) (f) for various (f, m). Particularly, for a node defined by f : (x, y) → xy, we prove that AD m(2) (xy) = m+1 4, and we deduce as a corollary that AD mfor any f, where mult 0 ∆ f is the multiplicity of the discriminant ∆ f at the origin in the deformation space. Furthermore, we show that the function m → AD m(2is an analytic invariant measuring how much the singularity "counts as" an inflection point. We obtain 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 various related enumerative problems. CONTENTSANAND P. PATEL AND ASHVIN A. SWAMINATHAN 6.3. Computation of AD 2 (1,1) (f) 50 6.4. Computation of AD 3 (2) (y t − x s ) 51 6.5. Computation of AD 2 (1) (y t − x s ) 55 6.6. Computation of AD 4 (2) (y 2 − x s ) 59 6.7. Computation of AD 3 (1) (y 2 − x s ) 62 6.8.

Jet Bundles on Gorenstein Curves and Applications

“…A different approach, via relative Hilbert schemes, has been taken by Ran [48]. In [46], the authors apply their theory of automatic degeneracies to give a new proof of Proposition 2.1. More precisely, after a suitable Chern class calculation, which we review below in the language of jet bundles, the authors subtract the individual contribution of each node in the pencil to get the desired answer.…”

“…This allows one to mimic the usual procedure adopted in the smooth category. Related constructions have recently been reconsidered by A. Patel and A. Swaminathan in [46], under the name of sheaves of invincible parts, motivated by the classical problem of counting hyperflexes in one-parameter families of plane curves. Besides loc.…”

“…In the first section we describe the construction of principal parts, jet bundles (with a glimpse on an abstract construction by Laksov and Thorup) and invincible parts by Patel and Swaminathan. In Section 2 we describe two applications of locally free replacements: the enumeration of hyperflexes in families of plane curves via automatic degeneracies [46], and the determination of the class of the stable hyperelliptic locus in genus 3 [19]. In Section 3 we define ramification points of linear systems on smooth curves; we introduce the classical Wronskian section attached to a linear system and state the associated Brill-Segre formula.…”

“…They are intended to deal with degenerations of ramification points of linear systems in one parameter families of curves of fixed arithmetic genus. In fact, in Section 2 we shall give two applications to see the theory in action: the count of hyperflexes in a pencil, as performed in [46], and the determination of the class of the stable hyperelliptic locus in genus 3, as worked out by Esteves [19].…”

“…Invincible parts. An elegant approach to the problem of locally free replacements of principal parts has been proposed by Patel and Swaminathan in their recent report [46]. Their construction is formally more adherent to the purely algebraic definition of principal parts as described in Section 1.3.…”