The enumerative geometry of cubic hypersurfaces: point and line conditions

Abstract

In order to count the number of smooth cubic hypersurfaces tangent to a prescribed number of lines and passing through a given number of points, we construct a compactification of their moduli space. We term the latter a 1-complete variety of cubic hypersurfaces in analogy to the space of complete quadrics. Paolo Aluffi explored the case of plane cubic curves. Starting from his work, we construct such a space in arbitrary dimension by a sequence of five blow-ups. The counting problem is then reduced to the computation of five Chern classes, climbing the sequence of blow-ups. Computing the last of these is difficult due to the fact that the vector bundle is not given explicitly. Identifying a restriction of this vector bundle, we arrive at the desired numbers in the case of cubic surfaces. ContentsA similar reasoning as in Lemma 2.5 shows also the following.Lemma 2.10. The liftc (i,i,j) = s 2 i s j for i, j > 1, c (i,i,i) = s 3 i for i ≠ 0, 1, c (i,j,k) = 2s i s j s k for k > j > i > 0.