Quartic surfaces with icosahedral symmetry

Abstract

We study smooth quartic surfaces in P 3 which admit a group of projective automorphisms isomorphic to the icosahedron group.1 This result was independently obtained by S. Mukai. 1 Proposition 2. Let S 4 W ∨ and S 4 U 4 be the fourth symmetric power of W 4 and of U 4 , and let () G denote the subspace of G-invariant elements. ThenThus we have two pencils of invariant quartic polynomials (S 4 W 4 ) A 5 and (S 4 U 4 ) 2.A 5 , so our quartic surface S is a member of one of them.

Finite collineation groups and birational rigidity

“…We claim that there are exactly 10 lines in P 3 that are common secants of both twisted cubic curves C 1 and C 2 (cf. [46, §2] or [19,Remark 2]). Indeed, denote by E 1 1 the π 1 -exceptional surface.…”

“…Using this description, we deduce that φ 1 (C 2 ) has 10 double points (cf. [19,Remark 5]), and these singular points form one Γ-orbit in P 2 . Note that the singular points of φ 1 (C 2 ) are ordinary double points.…”

Geometry of the Wiman–Edge pencil, I: algebro-geometric aspects

“…This plane sextic is also discussed in [9]. With the help of the Plücker formula (e.g., [8, formula 1.50]), it then follows that the dual curve of the Winger sextic is a Klein decimic.…”

K3 surfaces with maximal finite automorphism groups containing M 20

“…This was done by Kondo for the surface X Ko , we recall it here to have a complete picture, and we compute it for X Mu and X BH . Accordingly to [5,Section 3] the transcendental lattice of X Mu was already known by Mukai, but we could not find explicit computations, so we give it here. We give also equations for the three surfaces.…”

“…We give also equations for the three surfaces. Mukai already provided equations for X Mu as a smooth quartic surface in P 3 (C) (which is the Maschke surface, see [5,Section 3]) we compute it here in a different way, but we show that up to a projective transformation of P 3 (C), these are equivalent.…”