A228073 Degrees of special cubic divisors C_i. Each C_i corresponds to the collection of cubic fourfolds in the complex projective space of dimension 5, and these cubic fourfolds are indexed by i (i.e., of degree i). For each index i, the degree can be interpreted as the number of such cubic fourfolds.

-2, 0, 0, 192, 3402, 0, 196272, 917568, 0, 11327232, 32453136, 0, 200946468, 447835392, 0, 1873218816, 3568293162, 0, 11587373280, 19916839872, 0, 54185699328, 86195929680, 0, 205762215024, 309802123008, 0, 668839597248, 961259741172, 0, 1916372334240, 2662571684160, 0, 4975266781440, 6699605038272, 0, 11865484014132, 15620739902208, 0, 26444647324032, 34029871628112, 0, 55430090586720, 70205139813312, 0, 110613834846720, 137669694779232, 0

Offset

0,1

Comments

In the formula, alpha(q) and beta(q) are level three modular forms of weight 1 and 3 that generate the space of modular forms with respect to the group Gamma_0(3). Legendre(p/q) is the Legendre symbol.

Links

Table of n, a(n) for n=0..47.

B. Hassett, Special Cubic Fourfolds, Compositio Mathematica 120 (2000), no 1, 1-23.

L. Zhang and Z. Li, Modular Forms and Special Cubic Fourfolds, Advances in Mathematics, 2013, 315-326.

L. Zhang and Z. Li, Modular Forms and Special Cubic Fourfolds, arXiv:1203.1373 [math.AG], 2012-2015.

Formula

a(n) = coefficient of q^n in (-alpha3^11 + 162*alpha3^8*beta3 + 91854*alpha3^5*beta3^2 + 2204496*alpha3^2*beta3^3 - alpha^11 + 66*alpha^8*beta - 1386*alpha^5*beta^2 + 9072*alpha^2*beta^3), where alpha(q) = 1+6*Sum_{n>=1}(q^n * Sum_{d|n}Legendre(d/3)); alpha3(q) = alpha(q^3); beta(q) = Sum_{n>=1}(q^n * Sum_{d|n}(n/d)^2*Legendre(d/3)); beta3(q) = beta(q^3).

Example

For n = 0, a(0)= -2, corresponds to degree of special cubic fourfold of discriminant 0;
For n = 3, a(3) = 192, corresponds to degree of special cubic fourfold of discriminant 2*3 = 6;
For n = 4, a(4) = 3402, corresponds to degree of special cubic fourfold of discriminant 2*4 = 8;
In general, a(n) != 0 if and only if 2n==0,2 mod 6.

Crossrefs

Sequence in context: A293140 A008551 A183896 * A306936 A027652 A350022

Adjacent sequences: A228070 A228071 A228072 * A228074 A228075 A228076

Keyword

sign

Author

Letao Zhang, Aug 08 2013

Status

approved