A focus on focal surfaces

Abstract

Many classical problems in algebraic geometry have regained interest when techniques from differential geometry were introduced to study them. The modern foundations for this approach has been given by Griffiths and Harris in [6], who obtained in this way several classical and new results in algebraic geometry. More recently, this idea has been successfully followed by McCrory, Shifrin and Varley in [12] and [13] to study differential properties of hypersurfaces in P 3 and P 4 . In fact these two papers have greatly influenced the present work.In this spirit, the subject of this paper is the systematic study of focal surfaces of smooth congruences of lines in P 3 . This is indeed a clear example of a topic of differential nature in algebraic geometry. The study of such congruences has been very popular among classical algebraic geometers one century ago. Especially Fano has given many important contributions to this field. An essential ingredient in his work has been the focal surface of the congruence. This point of view has been retaken by modern algebraic geometers, such as Verra and Goldstein, and also by Ciliberto and Sernesi in higher dimension.What we find amazing in the papers by the classics is how much information they were able to provide about the focal surface of the known examples of congruences, in particular about its singular locus (and more especially about fundamental points). They seemed to have in mind some numerical relations that they never formulated explicitly. And even nowadays such kind of relations would require deep modern techniques, like multiple-point theory, but also this powerful machinery is not a priori enough since some generality conditions need to be satisfied.As a sample of this, the degree and class of the focal surface -the only invariants easy to compute-can be derived immediately from the Riemann-Hurwitz formula. However these invariants, even in the easiest examples (see Example 2.4 or Remarks after Corollary 4.7) seem to be wrong at a first glance. This is due to the existence of extra components of the focal surface or to the possibility that the focal surface counts with multiplicity, although this was never mentioned explicitly by the classics. Even in [5], these possibilities seem not to have been considered.The starting point of this work was to understand how the classics predicted the number of fundamental points of a congruence. We only know of one formula in the literature involving this number, which is however wrong (see Example 1.15 and the remark afterwards). So our first goal was to use modern techniques in order to rigorously obtain some of the classical results on the topic. Specifically, by regarding the focal surface of a congruence as a scheme, we reobtain its invariants (degree, class, class of its hyperplane 1 section, sectional genus, and degrees of the nodal and cuspidal curves) and give them a precise sense.We also restrict our attention to congruences of bisecants to a curve, or flexes to a surface (since they are special cases in the work by...

Line Congruences of Low Order

“…Observe that on a congruence of order zero or one both definitions coincide. We state now a very well-known result, of which we will give a brief intuitive geometric proof (a complete rigorous proof in nowadays language can be found for instance in [12] or [3] for n = 3; for arbitrary n the proof works in the same way as in [3], and in fact the same authors are preparing a paper containing in particular such a proof). Proof.…”

“…It is important to observe that the above degree is only valid as a scheme, since the focal locus can have non reduced components with some multiplicity (see [3] for some pathological examples). And we also point our attention that this is the degree as a hypersurface, i.e.…”

“…The projective bundle P(E) gives thus a desingularization of the subset of G (1, 3) consisting of all the lines meeting C. Hence there is a surfaceX ⊂ P(E) that is birational to X under the natural map π : P(E) → G (1, 3). On the other hand, our choice of the twist in E implies that the class t of the tautological line bundle O P(E) (1) of P(E) is the pull-back by π of the hyperplane section of G (1,3). Using that E is a vector bundle on P 1 we can write…”

“…These two tangent directions come from the intersection in P 5 of G (1,3) and the embedded tangent plane of X at L. It is well-known that the embedded tangent plane is contained in G(1, 3) for any line if and only if X is either the congruence of all the lines passing through a fixed point or the congruence of all the lines contained in a fixed plane (see for instance [12], Corollary 4.5.1). So that we can assume that the intersection of G (1, 3) with the embedded tangent plane at a general L ∈ X is always a degenerate conic.…”

“…We will then state and briefly prove the main properties of the focal locus of line congruences that we will need. A complete study of these (and many other) properties in the case of line congruences in P 3 (the main case treated in this paper) can be found in [3], while for an arbitrary P n the results will appear in a forthcoming paper of Marina Bertolini, Cristina Turrini and the author. Anyway most of the results of this section (and the next one) are very classical and has been reproved in different ways by many authors.…”