Intersection of ACM-curves in ℙ³

Abstract

In this note we address the problem of determining the maximum number of points of intersection of two arithmetically Cohen-Macaulay curves in P 3 . We give a sharp upper bound for the maximum number of points of intersection of two irreducible arithmetically Cohen-Macaulay curves C t and C tÀr in P 3 defined by the maximal minors of a t  ðt þ 1Þ, resp. ðt À rÞ Â ðt À r þ 1Þ, matrix with linear entries, provided C tÀr has no linear series of degree d c tÀrþ1 3 À Á and dimension n d t À r.

On the intersection of ACM curves in P3

“…Note: the existence of smooth ACM curves C s , C t and the computation of the intersection number was done by another method in [24]. The proof of the maximality is new here.…”

“…Giuffrida in [10] and Diaz in [5] proved that the number of intersection points of two smooth non-planar irreducible curves C 1 and C 2 in P 3 of degrees d 1 and d 2 , respectively, is bounded by (d 1 − 1)(d 2 − 1) + 1 and the maximum is reached only if C 1 and C 2 are both on the same quadric surface. And a result of the second author with Ranestad in [24] showed the existence of certain ACM curves with conjectured maximum order of intersection.…”

On the intersection of ACM curves in $\PP^3$

“…Proof. Note: the existence of smooth ACM curves C s , C t and the computation of the intersection number was done by another method in [24]. The proof of the maximality is new here.…”