Rational Curves in Calabi-Yau Threefolds

Rational Curves in Calabi-Yau Threefolds 3927
to a quartic hypersurface Q containing N 5 given 3-spaces in the Ffibration
of T. Then, for general such Q, we see that Q \ T is the union
of a Calabi-Yau threefold and the N 5 given 3-spaces. Each rational
curve C of type (d, a) for a 1 intersects each 3-space in at most a, and
hence a finite number of, points. For a given rational curve C 0 we want
to study the following subset
p 1
1 ð½C0ŠÞ ¼ fð½C0Š; ½FŠÞjC0 2 Fg
of the incidence I ¼ Id,a.
Finding the dimension of p 1
1 ([C0]) is essentially, as we shall see
below, equivalent to finding h 0 (J(4)) (or (h 1 (J(4))), where J is the ideal
sheaf in P N of the union X of C0 and the N 5 disjoint, linear 3-spaces,
each intersecting C0 as described. We then have the following result,
which is Corollary 1.9 of Sidman (2001):
Lemma 4.4. Let J be the ideal sheaf of a projective scheme X that consists
of the union of d schemes X1, ...,Xd in P N , whose pairwise intersections
are finite sets of points. Let mi be the regularity of Xi. Then J is
P d
i¼1 mi-regular.
We will use this result. Look at the following exact sequence:
0 ! J X=Tð4HÞ !OTð4HÞ !OX ð4HÞ !0:
This gives rise to the exact cohomology sequence
0 ! H 0 ðJX=Tð4HÞÞ ! H 0 ðOTð4HÞÞ
! H 0 ðOX ð4HÞÞ ! H 1 ðJX=Tð4HÞÞ ! 0:
This gives:
h 0 ðJX=Tð4HÞÞ ¼ h 1 ðJX=Tð4HÞÞ þ h 0 ðOTð4HÞÞ h 0 ðOX ð4HÞÞ
¼ h 1 ðJX=Tð4HÞÞ þ 35ðN 2Þ ð35ðN 5Þþ4d þ 1 ðN 5ÞaÞ
¼ h 1 ðJX=Tð4HÞÞ þ 105 ð4d þ 1 þð5 NÞaÞ:
Hence we see that
dim p 1 ð½C0ŠÞ ¼ h 1 ðJX=Tð4HÞÞ þ 104 dim Md;a
¼ h 1 ðJX=Tð4HÞÞ þ dim G dim Md;a;