Key varieties for prime Q-Fano threefolds defined by Jordan algebras of cubic forms. Part II

Subsequent to the previous paper [Tak5], we are concerned with the classification of complex prime Q-Fano $3$-folds of anti-canonical codimension 4 which are produced, as weighted complete intersections of appropriate weighted projectivizations of certain affine varieties related with P²²-fibrations. Such affine varieties or their appropriate weighted projectivizations (possibly allowing some coordinates have weights $0$) are called key varieties for prime Q-Fano 3-folds. The purpose of this paper is to give new constructions of a $14$-dimensional affine variety ΥA¹⁴ and a $15$-dimensional affine variety ΠA¹⁵ related with P²²-fibration, which were constructed and were shown to be key varieties in the papers [Tak3,Tak4] and [Tay]. It is well-known that the affine cone of the Segre embedded P²² is defined as the null loci of the so called -mapping of a 9-dimensional nondegenerate quadratic Jordan algebra J of a cubic form. Inspired with this fact, we construct ΥA¹⁴ and ΠA¹⁵ in the same way coordinatizing J with 9 and 10 parameters, respectively. The coordinatization with $9$ parameters is derived by using a fixed primitive idempotent, and the associated Peirce decomposition. The coordinatization with $10$ parameters is derived from the construction of a quadratic Jordan subalgebra generated by -products of two elements due to Petersson [Pe1].