Rational Curves in Calabi-Yau Threefolds

3928 Johnsen and Knutsen
if 4e4 (N 5) 2. If we work with a stratum W of M ¼ Md,a
where h 1 (JX=T(4H)) is constant, say c, then the incidence stratum
p 1 (W) has dimension c þ dim G codim(W, M). Now it is clear that
h 1 (J X=T(4H)) ¼ h 1 (J X=P N(4)) ¼ h 1 (J(4)), since h 1 (J T=P N (4)) ¼ 0, which
is true because rational normal scrolls are projectively normal. Moreover
h 1 (J(4)) ¼ 0ifX is 5-regular, by the definition of m-regularity in
general. By Theorem 1.1 of Gruson et al. (1983) we have:
Lemma 4.5. A non-degenerate, reduced, irreducible curve of degree d in P r
is (d þ 2 r)-regular.
Moreover in Corollary 1.10 of Sidman (2001) one has:
Lemma 4.6. The ideal sheaf of s linear k-spaces meeting (pairwise) in
finitely many (or no) points is s-regular.
Putting these two results together, we observe that if C0 spans an
r-space, then X is (d þ 2 r þ N 5) ¼ (d r þ N 3)-regular. In particular,
if C0 is a rational normal curve, then X is (N 3)-regular, and, in particular,
5-regular if N is 7 or 8 (and of course d N then). Also, curves
spanning a (d 1)-space are 5-regular if N ¼ 7 (for d 8). We then have:
Corollary 4.7. On a general F in T of type (1, 1, 1, 1) in P 7 there are only
finitely many smooth rational curves of degree at most 4. On a general F in
T of scroll type (2, 1, 1, 1) in P 8 there are only finitely many smooth
rational curves of degree at most 3.
Proof. We deduce that all X in question are 5-regular, so h 1 (J X=P N(4H)) ¼ 0
for all X, and hence all non-empty incidence varieties I d, a have dimension
equal to dim G, and hence the second projection map p2 has finite
fibres over general points of G. &
4.2. Analogous Questions for Other Threefolds
We would like to remark on the possibility of finding an analogue of
Theorem 4.3. Is it possible to produce isolated, rational curves of bidegree
(d, a) for many (d, a), also on general CY threefolds of intersection
type (2H c1F, 3H c2F) on five-dimensional rational normal scrolls
in P N ? Here we obviously look at fixed (c1, c2) such that c1 þ c2 ¼ N 6.
A natural strategy, analogous to that in the previous section,
would be to limit oneself to work with rational normal scrolls