The Automorphism Groups of Enriques Surfaces Covered by Symmetric Quartic Surfaces

Let S be the (minimal) Enriques surface obtained from the symmetric quartic surface ( P i<j xixj) 2 = kx1x2x3x4 in P with k 6= 0, 4, 36, by taking quotient of the Cremona action (xi) 7→ (1/xi). The automorphism group of S is a semi-direct product of a free product F of four involutions and the symmetric group S4. Up to action of F , there are exactly 29 elliptic pencils on S. The automorphism groups of very general Enriques surfaces, namely those corresponding to very general points in moduli, were computed in BarthPeters[1] as an explicitly described infinite arithmetic group. Also many authors [3, 1, 9, 5] studied Enriques surfaces with only finitely many automorphisms. The article [1] also includes an example whose automorphism group is infinite but still virtually abelian group. In this paper we give a concrete example of an Enriques surface whose automorphism group is not virtually abelian. Moreover, the automorphism group is explicitly described in terms of generators and relations. See also Remark 5. We work over any algebraically closed field whose characteristic is not two. Let us introduce the quartic surface with parameters k and l, (1) X : {s2 = ks4 + ls1s3} ⊂ P, where sd are the fundamental symmetric polynomials of degree d in the homogeneous coordinates x1, . . . , x4. It is singular at the four coordinate points (1 : 0 : 0 : 0), . . . , (0 : 0 : 0 : 1) and has an action of the symmetric group S4. It also admits the action of the standard Cremona transformation ε : (x1 : · · · : x4) 7→ ( 1 x1 : · · · : 1 x4 ) which commutes with S4. After taking the minimal resolution X, the quotient surface S = X/ε becomes an Enriques surface, whenever X avoids the eight fixed points (±1 : ±1 : ±1 : 1) of ε. This condition is equivalent to k + 16l ̸= 36, k ̸= 4 and 4l + k ̸= 0. 2000 Mathematics Subject Classification. 14J28, 20F55. Supported in part by the JSPS Grant-in-Aid for Scientific Research (B) 22340007, (S) 19104001, (S) 22224001, (S)25220701, (A) 22244003, for Exploratory Research 20654004 and for Young Scientists (B) 23740010.