Two Formulas for the Number of Lines on Complex Projective Hypersurfaces

Abstract

Two formulas for the classical number $C_n$ of lines on a generic hypersurface of degree $2n-3$ in $\mathbb{CP}^n$ are obtained which differ from the formulas by Dominici, Harris, Libgober, and van der Waerden-Zagier. We review the splitting principle computation by Harris obtaining a similar general closed-form formula in terms of the Catalan numbers and elementary symmetric polynomials. This in turn yields $C_n$ as a linear difference recursion relation of unbounded order. Thus, for the sequence of certain linear combinations of $C_n$, a simple generating function is found. Then, a result from random algebraic geometry by Basu, Lerario, Lundberg, and Peterson, that expresses these classical enumerative invariants as proportional to the Bombieri norm of particular polynomial determinants, yields another combinatorial expansion in terms of certain set compositions and block labeling counting. As an example, we compute this combinatorial interpretation for the cases of 27 lines on a cubic surface and 2875 lines on a quintic threefold. As an application, we reobtain the parity and asymptotic upper bound of the sequence. In an appendix, we generalize the splitting principle calculation to obtain a formula for the number of lines on a generic complete intersection.