The Application of the Method of Horace to Get Number of Generators for an Ideal of s General Points in P4

Abstract

Let S be a general set of s points in P4, and R the homogeneous coordinate ring of P4. Then the ideal of S, IS has a minimal free resolution of the form: 0 −−−→ F3 −−−→ F2 −−−→ F1 −−−→ F0 −−−→ IS −−−→ 0 where Fp = R(−d − p)ap−1

R(−d − p − 1)bp , d being the smallest integer satifying s ≤ h0(P4,OP4 (d)) and ap = h0(TS ⊗ Ωp+1 P4 (d+ p + 1)), bp = h1(TS ⊗ Ωp+1 P4 (d + p + 1)) and

d+3 4

< s ≤

d+4 4

, with 0 ≤ p ≤ 3 and when p = 0, we would have ap−1 =

d+4 4

− s and when p = 3 then bp = s−

d+3 4

. In this paper I prove that either a0 = 0 or b0 = 0 by proving maximal rank for the map: H0

ΩP4(d + 1)

−→ s i=1 ΩP4(d + 1)|Si by use of the methods of Horace to prove bijectivity for a specific number of fibres and then maximal rank for a general set.