Cox rings of cubic surfaces and Fano threefolds

Abstract

We determine the Cox rings of the minimal resolutions of cubic surfaces with at most rational double points, of blow ups of the projective plane at non-general configurations of six points and of three dimensional smooth Fano varieties of Picard numbers one and two.Comment: 34 pages, section two expanded. To appear in Journal of Algebr

Computing automorphisms of Mori dream spaces

“…8,9] for details. The Cox rings of singular cubic surfaces with at most ADE singularities have been determined in [7,6]. So far, the automorphism groups have been computed in the cases without parameters [15].…”

“…We use that R is effectively and pointedly graded by Z 5 : the degrees of the variables T i are the columns of  Applying Algorithm 3.8 to R, we obtain Stab I (Aut K (S)) ∼ = S 5 . Ten of the 120 elements are just permutations of variables: written as elements of S 10 , we have besides the identity (10,9,7,4,8,6,3,5 Then the computation of the Gröbner fan of I with gfan [13] using the symmetries option and the ten listed symmetries was computed instantly on a 5 year old machine (core2duo), whereas the case of no given symmetries took about two seconds.…”

“…, w 7 } with w i := deg(T i ). An application of Algorithm 3.7 shows that Aut K (S) consists of the matrices A, AB ∈ GL (7), where A is a Ω S -diagonal matrix and B a Aut(Ω S )-permuting matrix as follows:…”

“…Using Algorithm 3.8, we compute an ideal J ⊆ K[T i,j ; 1 ≤ i ≤ 7] describing Aut K (R) ⊆ GL (7); it is given by…”

“…5.5]. Then I(P ) describes H ⊆ CAut K (R) inside GL(k).The families of Cox rings of the resolutions of cubic surfaces have been determined in[7]. The Cox rings of the cubic surfaces are obtained by contracting all (−2)-curves according to [10, Prop.…”