On surfaces in P with no trisecant lines

Examples of surfaces in P6 with no trisecant lines are constructed. A classification recovering them is given and conjectured to be the complete one. Introduction The study of varieties embedded in P with no trisecant lines is a very classical problem in algebraic geometry. The simplest case, i.e. the case of space curves goes back to Castelnuovo. For surfaces the problem has been studied in codimension 2 and 3. In [2] Aure classifies smooth surfaces in P4 with no trisecant through the general point in the space. In [3] Bauer classifies smooth surfaces in P5 with no trisecant lines through the general point on the surface. Here we treat the case of smooth surfaces in P6 with no trisecant lines at all. Of course this includes all surfaces which are cut out by quadrics, but there are some more examples. We proceed as follows: • In the first two sections we construct examples of surfaces in P6 with no trisecant lines. Standard examples are used to construct new examples, via linkage. In each case we indicate whether the ideal is generated by quadrics or not. The cases of surfaces containing lines and surfaces with no lines are treated separately. • In section 3 we give a complete list of surfaces with no proper trisecant lines and no lines on it. • In section 4 we give a classification of surfaces with lines on the surface but no proper trisecant lines. These include scrolls, conic bundles, surfaces with an isolated (−1)-line and inner projections from P7, i.e. projections of smooth surfaces in P7 from a smooth point on the surface. The list of cases produced in sections 3 and 4 cover all the examples constructed in sections 1 and 2. Of course this does not cover all the possibilities. In fact, surfaces with a finite number of disjoint (−1)-lines need not be inner projections from P7. If in addition every (−1)-line meets some other line L of selfintersection L2 ≤ −2 on the surface our methods do not apply. However, checking the cases with less than one hundred (−1)-lines give no new examples, so we conjecture that the list is in fact complete. The main results of this work are summarized in the following: