Key varieties for prime $\mathbb{Q}$-Fano threefolds defined by Jordan algebras of cubic forms. Part I

Abstract

<jats:p> The main aim of this paper is to construct certain complex prime <jats:inline-formula> <jats:tex-math>\mathbb{Q}</jats:tex-math> </jats:inline-formula> -Fano <jats:inline-formula> <jats:tex-math>3</jats:tex-math> </jats:inline-formula> -folds anticanonically embedded of codimension <jats:inline-formula> <jats:tex-math>4</jats:tex-math> </jats:inline-formula> . We show that such <jats:inline-formula> <jats:tex-math>3</jats:tex-math> </jats:inline-formula> -folds can be obtained as weighted complete intersections of suitable weighted projectivizations of a <jats:inline-formula> <jats:tex-math>13</jats:tex-math> </jats:inline-formula> -dimensional affine variety <jats:inline-formula> <jats:tex-math>\mathscr{H}_{\mathbb{A}}^{13}</jats:tex-math> </jats:inline-formula> and its related varieties, which we call key varieties. More precisely, we prove that a prime <jats:inline-formula> <jats:tex-math>\mathbb{Q}</jats:tex-math> </jats:inline-formula> -Fano <jats:inline-formula> <jats:tex-math>3</jats:tex-math> </jats:inline-formula> -fold of genus <jats:inline-formula> <jats:tex-math>3</jats:tex-math> </jats:inline-formula> with three <jats:inline-formula> <jats:tex-math>1/2(1,1,1)</jats:tex-math> </jats:inline-formula> -singularities in the family No. 5.4 of Takagi (2002) arises as a linear section of a weighted projectivization of <jats:inline-formula> <jats:tex-math>\mathscr{H}_{\mathbb{A}}^{13}</jats:tex-math> </jats:inline-formula> , and conversely, any such <jats:inline-formula> <jats:tex-math>3</jats:tex-math> </jats:inline-formula> -fold can be obtained in this way. Furthermore, by relating <jats:inline-formula> <jats:tex-math>\mathscr{H}_{\mathbb{A}}^{13}</jats:tex-math> </jats:inline-formula> to the <jats:inline-formula> <jats:tex-math>C_{2}</jats:tex-math> </jats:inline-formula> -cluster variety constructed by Coughlan and Ducat (2020), we recover their constructions of prime <jats:inline-formula> <jats:tex-math>\mathbb{Q}</jats:tex-math> </jats:inline-formula> -Fano <jats:inline-formula> <jats:tex-math>3</jats:tex-math> </jats:inline-formula> -folds belonging to <jats:inline-formula> <jats:tex-math>108</jats:tex-math> </jats:inline-formula> families in the Graded Ring Database. The variety <jats:inline-formula> <jats:tex-math>\mathscr{H}_{\mathbb{A}}^{13}</jats:tex-math> </jats:inline-formula> is constructed by adapting the description of the affine cone over the Segre variety <jats:inline-formula> <jats:tex-math>\mathbb{P}^{2}\times \mathbb{P}^{2}</jats:tex-math> </jats:inline-formula> , defined as the null locus of the <jats:inline-formula> <jats:tex-math>\sharp</jats:tex-math> </jats:inline-formula> -mapping of a <jats:inline-formula> <jats:tex-math>9</jats:tex-math> </jats:inline-formula> -dimensional quadratic Jordan algebra <jats:inline-formula> <jats:tex-math>J</jats:tex-math> </jats:inline-formula> of a cubic form, to a coordinatization of <jats:inline-formula> <jats:tex-math>J</jats:tex-math> </jats:inline-formula> with <jats:inline-formula> <jats:tex-math>8</jats:tex-math> </jats:inline-formula> parameters derived via a Peirce decomposition with respect to three fixed complementary primitive idempotents. </jats:p>