Rational Curves in Calabi-Yau Threefolds
Rational Curves in Calabi-Yau Threefolds
Rational Curves in Calabi-Yau Threefolds
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<strong>Rational</strong> <strong>Curves</strong> <strong>in</strong> <strong>Calabi</strong>-<strong>Yau</strong> <strong>Threefolds</strong> 3933<br />
def<strong>in</strong>ed by jLj and pick a subpencil fDlg jDj ’P 1 2D2þ1 generated by<br />
two smooth curves (so that, <strong>in</strong> particular, fDlg is without fixed components,<br />
and with exactly D 2 base po<strong>in</strong>ts). Each fL(Dl) will span a<br />
(h 0 (L) h 0 (L D) 1)-dimensional subspace of P g , which is called the<br />
l<strong>in</strong>ear span of fL(Dl) and denoted by Dl. Note that Dl ¼ P cþ1þ1 2D2 .<br />
The variety swept out by these l<strong>in</strong>ear spaces,<br />
T ¼ [<br />
Dl P g ;<br />
l2P 1<br />
is a rational normal scroll (see Schreyer, 1986) of type (e1, ..., ed), where<br />
with<br />
ei ¼ #fjjdj ig 1; ð1Þ<br />
d ¼ d0 :¼ h 0 ðLÞ h 0 ðL DÞ;<br />
d1 :¼ h 0 ðL DÞ h 0 ðL 2DÞ;<br />
.<br />
.<br />
di :¼ h 0 ðL iDÞ h 0 ðL ði þ 1ÞDÞ;<br />
.<br />
Furthermore, T has dimension dim T ¼ d0 ¼ h 0 ðLÞ h 0 ðFÞ ¼cþ<br />
2 þ 1 2 D2 and degree deg T ¼ h 0 ðFÞ ¼g c 1 1 2 D 2 .<br />
We will need the follow<strong>in</strong>g.<br />
Lemma 5.2. Assume that L is ample, D 2 ¼ 0, and h 1 (L iD) ¼ 0 for all<br />
i 0 such that L iD 0. Then the scroll T def<strong>in</strong>ed by jDj as described<br />
above is smooth and of maximally balanced scroll type. Furthermore,<br />
dim T ¼ c þ 2 and deg T ¼ g c 1.<br />
Proof. Let r :¼ maxfijL iD 0g. Then by Riemann-Roch, and our<br />
hypothesis that h 1 (L<br />
f<strong>in</strong>ds r ¼b<br />
iD) ¼ 0 for all i 0 such that L iD 0, one easily<br />
g<br />
cþ2c and<br />
d0 ¼ ¼dr 1 ¼ L:D ¼ c þ 2;<br />
1 dr ¼ g þ 1 ðc þ 2Þr c þ 2;<br />
di ¼ 0 for i r þ 1;<br />
whence the scroll T is smooth and of maximally balanced scroll type. The<br />
assertions about its dimension and degree are immediate. &