Formulas for the Number of Lines on Complex Projective Hypersurfaces

Two formulas for the classical number Cₙ of lines on a generic hypersurface of degree 2n-3 in CPⁿ are obtained which substantially differ from Zagier's formula. Schubert calculus leads to an explicit general closed-form formula in terms of binomial coefficients, the Catalan numbers, and elementary symmetric polynomials evaluated at certain integers. This in turn yields Cₙ as a linear difference recursion relation of unbounded order. Thus, for the sequence of certain linear combinations of Cₙ, 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 counting. As an example, we compute the 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 without using Zagier's formula. In consequence, we provide two alternative expressions to the latter for the number of lines in hypersurfaces which may open new directions to study the arithmetic, combinatorial, and analytical properties of this enumerative geometry sequence.