Bounds for the Genus of Space Curves

Abstract

We compute the following upper bounds for the maximal arithmetic genus P,(d, t ) over all locally Cohen -Macaulay space curves of degree d, which are not contained in a surface of degree t -1: These bounds are sharp for t 5 4 and any d 2 t.

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…”

“…Theorem 1.1 ( [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…”

Intersection of ACM-curves in ℙ3

“…[1]) and considerably lower than the genus bound for locally Cohen-Macaulay curves not on surfaces of degree less than t (cf. [2]). Using the construction given in section 3, we prove the existence of irreducible arithmetically Cohen-Macaulay curves C t and C t−r in P 3 which meet in the conjectured maximum number of points.…”

2-Nilpotent co-Higgs structures

“…i − i) + α = dt − (d − 2)(d − 3)/2 + α;see[9,10,16]; for locally Cohen-Macaulay space curves, one can also use[3]. If p(t) is the Hilbert polynomial of the scheme Z, then we have χ(O Z ) = p(0) and so−(d − 2)(d − 3)/2 ≤ −2d, i.e.…”