Involutions on surfaces withp g =q = 1

Abstract

In this paper some numerical restrictions for surfaces with an involution are obtained. These formulas are used to study surfaces of general type S with p g = q = 1 having an involution i such that S/i is a non-ruled surface and such that the bicanonical map of S is not composed with i. A complete list of possibilities is given and several new examples are constructed, as bidouble covers of surfaces. In particular the first example of a minimal surface of general type with p g = q = 1 and K 2 = 7 having birational bicanonical map is obtained.

Some bidouble planes with pg= q = 0 and 4 ≤ K2≤ 7

“…This is similar to the proof of [18,. Notice that p g (P ) = q(P ) = 0 (because p g (P ) ≤ p g (S), q(P ) ≤ q(S)).…”

“…Theorem 3.1 (cf. [18]). Let S be a smooth minimal surface of general type with p g = 0 having an involution i such that the bicanonical map of S is not composed with i and S/i is not rational.…”

Involutions on surfaces with p g = q = 0 and K 2 = 3

“…Then the number of isolated fixed points of i is t = 5 and, if S/i is not rational, one of the following holds: Proof Proposition 3 (c) of [23] gives…”

Mixed Quasi-Étale Quotients With Arbitrary Singularities

“…Note that there is a surface with K 2 S = 6 and g alg = 7; to the best of our knowledge, this is the first example of a minimal surface of general type with p g = q = 1 and g alb > K 2 S ; we recall that this is not possible for K 2 S ≤ 3 by the classification [18][19][20]22]. Also, other examples with 4 ≤ K 2 ≤ 6 may be, to the best of our knowledge, new, although other surfaces with those invariants have been already constructed (see [28,30,33,[36][37][38]). Also, other examples with 4 ≤ K 2 ≤ 6 may be, to the best of our knowledge, new, although other surfaces with those invariants have been already constructed (see [28,30,33,[36][37][38]).…”