In this paper some numerical restrictions for surfaces with an involution are obtained. These formulas are used to study surfaces of general typeS withp _g =q = 1 having an involutioni such thatS/i is a nonruled surface and such that the bicanonical map ofS is not composed withi. 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 withp _g =q = 1 andK ² = 7 having birational bicanonical map is obtained.

中文翻译：

---

pg = q = 1的曲面上的对合

在本文中，获得了对合曲面的一些数值限制。这些公式被用于一般类型的研究表面小号与p _克= q = 1具有对合我使得S / I是一种nonruled表面和使得所述bicanonical地图小号不与由我。给出了可能性的完整列表，并构造了几个新的示例，作为曲面的双重覆盖。特别地，获得具有双理性双正则图的p _g＝q＝ 1且K ²＝ 7的一般类型的最小表面的第一示例。