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

The first aim of this paper is to construct a $13$-dimensional affine variety HA¹³ related with P²²-fibration. 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 HA¹³ in the same way coordinatizing J with 8 parameters. We derive such a coordinatization of J using fixed three complementary primitive idempotents and the associated Peirce decomposition of J. The second aim of this paper is to construct complex prime Q-Fano $3$-folds of anti-canonical codimension 4 as weighted complete intersections of appropriate weighted projectivizations of HA¹³ or its subvarieties (possibly allowing some coordinates with weight $0$). The affine variety HA¹³ and such weighted projectivizations of HA¹³ are called key varieties for prime Q-Fano 3-folds. We show that a prime Q-Fano 3-fold of genus $3$ with three 1/2(1,1,1)-singularities belonging to the class of No.5.4 as in [Tak1] is constructed from one weighted projectivization of HA¹³ such that all the coordinates have positive weights. Conversely, we also show that any such a prime Q-Fano 3-fold is obtained in this way. Moreover, relating HA¹³ with the C₂-cluster variety constructed by Coughlan and Ducat [CD1], we show that weighted projectivizations of HA¹³ or its subvarieties are key varieties for prime Q-Fano 3-folds belonging to 108 classes in the online database [GRDB].