Threefolds in ℙ5 with a 3-dimensional family of plane curveswith a 3-dimensional family of plane curves

Abstract

A classification theorem is given of smooth threefolds of P 5 covered by a family of dimension at least three of plane integral curves of degree d 2. It is shown that for such a threefold X there are two possibilities:(1) X is any threefold contained in a hyperquadric;(2) d 3 and X is either the Bordiga or the Palatini scroll.1.-Preliminaries and threefolds not of isolated type.Let X ⊂ P 5 be an integral projective variety of dimension 3, and degree d. We will always assume that X is non-degenerate, i.e. it is not contained in any hyperplane.We suppose that X contains an algebraic family F of dimension at least 3 of plane, integral curves. It is immediate to remark that, if the union of the curves of F does not cover X, then this union is a surface containing a 3-dimensional family of plane curves, hence a union of surfaces of P 3 . We will always exclude this situation, so from now on we assume that the curves of F cover X. Our aim is to classify such varieties X.If the curves of F are lines, then the answer is classical and is given by the following theorem.

Systems of conservation laws of Temple class, equations of associativity and linear¶congruences in P 4

“…When n ≥ 4 these congruences are not necessarily linear. In this case the focal varieties must be singular, as follows from [17].…”

“…In this subsection we discuss equations (17) and (18). Since both equations possess only one Riemann invariant, the corresponding focal varieties will be reducible, consisting of a cubic scroll and a plane intersecting the cubic scroll along its directrix.…”

“…In the case n = 4 the geometry of focal varieties of general linear congruences, known as the Palatini scrolls, was investigated in [22] (see also [17] and [21] for further properties of the Palatini scrolls) (we thank F. Zak for providing these references). T-systems of conservation laws corresponding to general linear congruences will be discussed elsewhere.…”

“…3). Equation (17) was discussed in [13] and [27]. The classification of third order equations of Monge-Amperé type was given in [1].…”

INTEGRABLE FOUR-COMPONENT SYSTEMS OF CONSERVATION LAWS AND LINEAR CONGRUENCES IN ${\mathbb P}^5$

“…Hence, this map defines the structure of a ‫ސ‬ 1 -bundle over V A on the jump locus (15). In the case n = 4 the geometry of focal varieties of linear congruences, also known as the Palatini scrolls, was investigated in [15] (see also [12], [14] and [19] for further properties of the Palatini scrolls).…”