Key varieties for prime Q-Fano threefolds defined by Freudenthal triple systems

In this paper, we are concerned with the classification of complex primeQ-Fano 3-foldsof anti-canonical codimension 4 which are produced, as weighted complete intersections ofappropriate weighted projectivizations of certain affine varieties related withP(1)xP(1)xP(1)-fibrations. Such affine varieties or their appropriate weighted projectivizations are called keyvarieties for primeQ-Fano 3-folds. We realize that the equations of the key varieties can bedescribed conceptually by Freudenthal triple systems (FTS, for short). The paper consistsof two parts. In Part 1, we revisit the general theory of FTS; the main purpose of Part 1 isto derive the conditions of so called strictly regular elements in FTS so as to fit with ourdescription of key varieties. Then, in Part 2, we define several key varieties for primeQ-Fano3-folds from the conditions of strictly regular elements in FTS. Among other things obtainedin Part 2, we show that there exists a 14-dimensional factorial affine varietyU(A)(14)of codimen-sion 4 in an affine 18-space with only Gorenstein terminal singularities, and we constructexamples of primeQ-Fano 3-folds of No.20544 in as reported by Alt & imath;nok et al. (The gradedring database,http://www.grdb.co.uk/forms/fano3) as weighted complete intersections ofthe weighted projectivization ofU(A)(14)in the weighted projective spaceP(1(15),2(2),3).Wealsoclarify in Part 2 a relation betweenU(A)(14)and theG(2)((4))-cluster variety, which is a key variety forprimeQ-Fano 3-folds constructed in Coughlan and Ducat (Compos. Math. 156:1873-1914,2020).