ALEXANDRU DIMCA A GEOMETRIC APPROACH TO THE CLASSIFICATION OF PENCILS OF QUADRICS Let Q be the vector space of quadratic forms q : C" --, C and denote by P(Q) the associated projective space. To a pencil of quadrics ~ :2ql +/~qe (q~ Q \ {0} and linearly independent) correspond two geometric objects: a line L in P(Q) and an algebraic variety V c P"- 1, the intersection of the two quadrics qi = 0. On the other hand, the classification of pencils of quadrics has been known for a long time and involves some intricate algebraic considerations (see [4, XIII] ). In this paper we show how the class of a pencil ~ is uniquely determined by the position of the line L with respect to the subvarieties in P(Q) defined by the rank of a quadratic form and by the singular part ZV of the variety V. In particular, the paper contains a complete description of this singular set Z V in terms of the Segre symbol of the pencil. It also contains the description of an associated sheaf on the line L similar to the theta-characteristic of a net of quadrics [i]. Our proofs depend on the algebraic classification (which allows us to write a pencil in a canonical form) but we hope that the geometric point of view brings more light on this classification and sometimes can be more effective in explicit situations. We would like to thank the referee for suggesting the use of sheaves associ- ated to a pencil. 1. SOME PROJECTIVE VARIETIES In this section we introduce and study some projective varieties which occur in the investigation of pencils of quadrics. Let W c P(Q) be the set of all quadratic forms of rank r, r = 1 .... , n. These are obviously irreducible smooth quasi-projective varieties and more- over we have the following. PROPOSITION 1.1 (i) wr: wuw_,u...~w,. (ii) codim W = (n - r)(n - r + 1)/2. (iii) For any 1 < r < n, IYV _ 1 is precisely the singular part of the variety 17¢. Proof. The first point is clear. To go further note that there is a natural action Gl(n) x Q -* Q whose orbits are the sets V = Gl(n)'qr, where qr = Geometriae Dedicata 14 (1983) 105 111. 0046 5755/83/0142 0105501.05. (~ 1983 by D. Reidel Publishing Company.