The existence of elliptic fibre space structures on Calabi–Yau threefolds, II

Abstract

In [11], we considered the question of existence of elliptic fibre space structures on a smooth Calabi–Yau threefold X. Necessary conditions are that there exists a nef divisor D on X with D³=0; D²[nequiv ]0 and D·c₂[ges ]0. In the case when D·c₂[ges ]0, it was observed that a result from [9] implies that there is an elliptic fibre space structure determined by D, and so [11] concerned itself with the case when D·c₂=0. In this case, we were not able to deduce in general the existence of an elliptic fibre space structure, but only in the presence of a further condition. In particular, it follows from the Theorem in [11] that, if r denotes the number of rational surfaces E on X with D[mid ]_E≡0, and if the Euler characteristic e(X)≠2r, then some positive multiple of D determines an elliptic fibre space structure on X (the theorem in fact proves a slightly stronger result, which gives the elliptic fibre space structure if there is any non-rational surface E on X with D[mid ]_E≡0). The purpose of this note is to clarify the meaning of the above rather mysterious condition on the non-vanishing of e(X)/2−r.