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.