Rational Curves in Calabi-Yau Threefolds

3950 Johnsen and Knutsen
projective space P N spanned by T. Study the incidence I ¼ Id, a ¼f([C],
[F]) 2 M G j C Fg. Then one easily shows:
Lemma 6.5. Every component of I has dimension at least dim G, and
dim G ¼ 104 if 4e 4 (N 5) 2.
Proof. Let t, u, Z1, ..., Z4 as usual be coordinates of T. Let r, s be
homogeneous coordinates of the C ¼ P 1 which is mapped into T as a
rational curve of bidegree (d, a). This map corresponds to some (not
uniquely defined) parametrization
t ¼ Tðr; sÞ; u ¼ Uðr; sÞ; Z1 ¼ S1ðr; sÞ; ...; Z4 ¼ S4ðr; sÞ:
Here deg T ¼ deg U ¼ a, and deg Si ¼ d aei, for i ¼ 1, 2, 3, 4. (A set of
coordinates for P n are of the form Z j
i ¼ Zit j u ei j
. Each coordinate is then
of degree d in the variables r, s. See for example Stevens (2000, p. 3), or
Reid (1997). The 6-tuples
ðTðr; sÞ; Uðr; sÞ; S1ðr; sÞ; ...; S4ðr; sÞÞ
depend on 2(a þ 1) þ (d ae1 þ 1) þ þ(d ae4 þ 1) ¼ 4d þ a(5 N) þ 6
variables. Let N ¼ Nd, a be the set of such 6-tuples. (An open subset of)
N can be viewed as a parameter space for ‘‘Parametrized rational curves
of bidegree (d, a) inT’’. The parameter space M can be viewed as a quotient
of N. Likewise J ¼ J d, a is defined as the corresponding incidence
in N G, and can be viewed as a quotient of I. The fibres of these quotients
have dimension 5 ¼ dim PGL(2) þ 2 (We have two multiplicative
factors, one for (T, U), and one for (S1, ..., S4)). Hence dim M ¼ 4d þ
a(5 N) þ 1, and dim I ¼ 4d þ a(5 N) þ 1 þ dim G ( ¼ 4d þ a(5 N) þ
105 if T is of reasonably well balanced type). Study the incidence
R ¼f(P, [F ]) 2 T GjP 2 F g. Then R has an equation, which is setting
equal to zero a sum of monomials of type:
pi1;...;i4ðt; uÞZi1 1
Z i4
4 :
The dim G þ 1 coefficients of the p i1, ..., i4 can be viewed as homogeneous
coordinates of G. In this equation we now insert the parametrizations
T(r, s), U(r, s), S1(r, s), ..., S4(r, s). This gives an equation involving the
coordinates of G and the dim N coefficients of these 6-tuples, and in
addition r, s. We may view this as a homogeneous polynomial of degree
4d þ a(5 N) inr, s, since deg pi 1, ..., i4(t, u) ¼ i1e1 þ þi4e4 (N 5).
The equation of the incidence J in N G is obtained by setting all the
4d þ a(5 N) þ 1 coefficients of this polynomial equal to zero. Since this