On the geometry of canonical curves of odd genus

Abstract

Fix an algebrically closed field k with char(k) = 0. Let C be a projective nonsingular curve of genus 5 defined over k, and let W 1 4 ⊂ Pic (C) be the subscheme of divisor classes of degree 4 and dimension 1. W 1 4 is a curve, which is irreducible and nonsingular of genus 11 if C is general, and can be identified with the singular locus of the theta divisorW4 ⊂ Pic(C). Under the Gauss rational map W 1 4 is mapped 2–1 onto a curve Γ ⊂ IP 4 which is nonsingular of degree 10 and genus 6. Γ is the set of vertices of the rank four quadrics containing the canonical image κ(C) ⊂ IP . The planes contained in the quadric QL, L ∈ W 1 4 , form the congruence (2-dimensional family) of 4-secant planes to κ(C). On each plane π of the congruence we have the set (κ(C) ∪ Γ) ∩ π consisting of 5 points. For a general choice of π these points are distinct and contained in a unique conic F ⊂ π which can be obtained as the locus of first order foci of the congruence, whereas the points ( κ(C) ∪ Γ ) ∩ π are the second order foci (see below for the definitions of first and second order foci). This geometrical configuration can be viewed as the first case of a whole series in two different ways. Firstly we can consider a sufficiently general curve C of genus g ≥ 5, and the locus W 1 g−1 ⊂ Pic g−1(C), which can be identified with the singular locus of the theta divisor Wg−1. W 1 g−1 has pure dimension g − 4; the projectivized tangent space at a point L ∈W 1 g−1\W 2 g−1 is the linear space vL ⊂ IP g−1 of dimension g− 5 vertex of the quadric QL of rank four containing the canonical curve κ(C), which is the projectivized tangent cone to Wg−1 at L. Each quadric QL has two rulings of IP g−3’s, and when L varies in W 1 g−1\W 2 g−1 they form a family of dimension g − 3. On each sufficiently general IP g−3 of the family there is a rational normal curve of first order foci. It has been shown in [CS] that the curve C can be recovered from this family, and this has been used to give a proof of Torelli’s theorem. Another extension of the genus 5 configuration can be introduced naturally, and it is the object of the present paper. Let’s consider, for any odd g = 2n+1, n ≥ 2, a sufficiently general projective irreducible nonsingular curve C of genus g, and the locusW 1 n+2 ⊂ Pic (C). By Brill-Noether theory it is known that W 1 n+2 is a nonsingular irreducible curve, whose genus has been computed by Kempf [K2] and by Pirola [P] (g = 11 for n = 2). For each L ∈W 1 n+2 the projectivized tangent cone to Wn+2 at L is a variety XL ⊂ IP , of degree n and dimension n+ 1, and XL = ⋃