A software package to compute automorphisms of graded algebras

Abstract

We present autgradalg.lib, a Singular library to compute automorphisms of integral, finitely generated ‫-ރ‬algebras that are graded pointedly by a finitely generated abelian group. The library implements algorithms of Hausen, Keicher and Wolf (Math. Comp. 86 (2017), 2955-2974. We apply these to Mori dream spaces and investigate the automorphism groups of a series of Fano varieties.

On smooth Fano fourfolds of Picard number two

“…The only serious task left open by Proposition 8.1 for explicitly computing h 1 .X; T X / is to determine the dimension of Aut.X /. As general tools, we mention Theorem 4.4 in [23], the algorithms presented thereafter and their implementation provided by [28]. The subsequent example discussions indicate how one might proceed in concrete cases.…”

“…For the varieties X from No. 1, the algorithm [28] is feasible and tells us that Aut.X / is of dimension 12. In particular, we see that also these varieties are infinitesimally rigid: In suitable linear coordinates respecting the grading, g D T 1 T 5 C T 2 T 6 C T 3 T 7 holds and the automorphisms on X are induced by the five-dimensional diagonally acting torus respecting g and the group GL.3/ acting on R.X / w 1 ˚R.X / w 5 via A .T 1 ; T 2 ; T 3 ; T 4 I T 5 ; T 6 ; T 7 / WD .A .T 1 ; T 2 ; T 3 /; T 4 I .A 1 / t .T 5 ; T 6 ; T 7 //:…”

“…Coates, Kasprzyk and Prince classified in [10] the smooth Fano fourfolds that arise as complete intersections of ample divisors in smooth toric Fano varieties of dimension at most eight. Comparing anticanonical self-intersection numbers as well as the first six coefficients of the Hilbert series yields that at least the 17 families 14, 15,24,25,28,30,32,33,38,44,45,46,47,51,52, 57 and 58 of Theorem 1.1 do not show up in [10].…”