A New Proof of a Theorem of Beorchia on the Genus of Space Curves

Abstract

We give a new proof of a theorem of BEORCHIA which provides a bound P ( d , t ) for t.he maximum genus of a locally Cohen-Macaulay space curve of degree d which does not lie on a nurface of degree t -1. zero. It is not known if this bound is sharp.BEORCHIA'S proof depends on a result of STRANO which holds only in characteristic zero. In this paper we give a new proof of BEORCHIA'S bound which is characteristic free. We follow the approach of [7], and the main ingredient of our proof is the results of [6] on the spectrum of a torsion free sheaf on P3, which impose restrictions on the speciality of a curve (there is an unnecessary characteristic zero assumption in [S]).The other ingredient is Proposition 3.2 below which follows from Halphen Speciality 141. ~ 1991 Mathematics Subject Classifidion. 14H50. Keyuords and phruses. Locally Cohen -Macaulay, space c u m , arithmetic genus.

A speciality theorem for Cohen-Macaulay space curves

“…The determinant trick (cf. [14]) shows that s(C) is less than or equal to the number of generators of A C over k [x, y, z]. Thus it is enough to show that the number of generators of A C over k [x, y, z] is at most deg C − m (e + 3 − m).…”

“…In the second case we have n ≥ e + 4, so that the right hand side max{0, deg C − n (e + 4 − n)} + n is at least deg C which is always an upper bound for s(C). [14]. These curves are interesting because of the maximum genus problem [14], and among them (when s = 2) are the extremal curves of [10,9] which have the largest Rao function among curves with fixed degree and genus.…”

The Maximum Genus Problem for Locally Cohen-Macaulay Space Curves

“…This paper is concerned with the problem of maximum genus for locally Cohen-Macaulay space curves: determine the maximum arithmetic genus of a locally Cohen-Macaulay space curve of degree d that is not contained in a surface of degree s − 1. The first author [6] proved a bound P (d, s) for the maximum genus if the characteristic of the ground field is zero, and proved the bound is sharp if s ≤ 4; later the third author [23] gave a different proof of this bound valid in any characteristic. Note that there is a locally Cohen-Macaulay curve of degree d that is not contained in a surface of degree < s if and only if d ≥ s ≥ 1.…”

“…It is clear that one must have d ≥ s because a curve is contained in the surface obtained as cone over a general plane section, while, if d ≥ s, an example of a curve of degree d not contained in a surface of degree < s is the divisor C = dL on S where L is a line contained in a smooth surface S of degree s. 6,23]). Let C be a curve in P 3 of degree d and genus g. Assume that C is not contained in any surface of degree < s. Then d ≥ s and…”

Smooth curves specialize to extremal curves

“…It is well known -and easy to show -that h 1 (O C (n)) = 0 if n ≥ d − 2: this is proven in characteristic zero in [19, Corollaire 2.4. (1)], and in any characteristic for example in [29,Proposition 3…”