A focus on focal surfaces.

*(English)*Zbl 1027.14024Congruences of lines in \(\mathbb P^3\), i.e. two-dimensional families of lines, and their focal surfaces, have been a popular object of study in classical algebraic geometry. They have been considered recently by several authors as Arrondo, Goldstein, Sols, Verra. Aim of the paper under review is to study from the modern point of view the notions appearing in this context, so to prove in a rigorous way some classical results.

More precisely, a congruence is a surface \(X\subset \mathbb G(1,3)\), its focal locus \(F\), in general a surface, is the branch locus of the natural projection \(q_X\) from the incidence correspondence \(I_X\) to \(\mathbb P^3\). It results that a general line of \(X\) contains two foci counted with multiplicity, a line \(L\) whose points are all foci is called a focal line.

In the paper under review the authors compute several invariants of the focal surface of a smooth congruence, e.g. its degree, class and sectional genus, and the degree of its nodal and cuspidal curves (assuming that these are the only 1-dimensional components of its singular locus). Then they study in detail some important examples of congruences, i.e. the bisecant lines of a smooth curve \(C\) in \(\mathbb P^3\), the bitangents and the inflectional tangents of a smooth surface \(\Sigma\). In the first case they find that in general the focal surface \(F\) is all formed by focal lines, which are the stationary bisecants. In the other examples they find, among other results, that \(\Sigma\) is a component of the focal surface with high multiplicity, and that at least one component of \(F\) is formed by focal lines.

Motivated by the examples they state a series of conjectures about congruences whose focal surface is not irreducible or not reduced, and about singularities of congruences of bitangents or inflectionary tangents to not necessarily smooth surfaces of \(\mathbb P^3\).

More precisely, a congruence is a surface \(X\subset \mathbb G(1,3)\), its focal locus \(F\), in general a surface, is the branch locus of the natural projection \(q_X\) from the incidence correspondence \(I_X\) to \(\mathbb P^3\). It results that a general line of \(X\) contains two foci counted with multiplicity, a line \(L\) whose points are all foci is called a focal line.

In the paper under review the authors compute several invariants of the focal surface of a smooth congruence, e.g. its degree, class and sectional genus, and the degree of its nodal and cuspidal curves (assuming that these are the only 1-dimensional components of its singular locus). Then they study in detail some important examples of congruences, i.e. the bisecant lines of a smooth curve \(C\) in \(\mathbb P^3\), the bitangents and the inflectional tangents of a smooth surface \(\Sigma\). In the first case they find that in general the focal surface \(F\) is all formed by focal lines, which are the stationary bisecants. In the other examples they find, among other results, that \(\Sigma\) is a component of the focal surface with high multiplicity, and that at least one component of \(F\) is formed by focal lines.

Motivated by the examples they state a series of conjectures about congruences whose focal surface is not irreducible or not reduced, and about singularities of congruences of bitangents or inflectionary tangents to not necessarily smooth surfaces of \(\mathbb P^3\).

Reviewer: Emilia Mezzetti (Trieste)

##### MSC:

14N05 | Projective techniques in algebraic geometry |

14M15 | Grassmannians, Schubert varieties, flag manifolds |

51N15 | Projective analytic geometry |