Congruences of lines in $${{\mathbb P}^5}$$ , quadratic normality, and completely exceptional Monge–Ampère equations

Abstract

The existence is proved of two new families of sextic threefolds in P 5 , which are not quadratically normal. These threefolds arise naturally in the realm of first order congruences of lines as focal loci and in the study of the completely exceptional Monge-Ampère equations. One of these families comes from a smooth congruence of multidegree (1, 3, 3) which is a smooth Fano fourfold of index two and genus 9.

Systems of conservation laws with third-order Hamiltonian structures

“…whose algebraic meaning was clarified in [16]. An important invariant of Hamiltonian operator (3) is its singular variety, det g ij = 0, which, due to (7), is a double hypersurface of degree n − 1:…”

“…In particular, reciprocally related systems (1) correspond to projectively equivalent congruences (2). Algebro-geometric aspects of this correspondence were investigated in [6,7,8,26]. In this paper we utilise the above geometric correspondence for the classification of systems (1) possessing third-order Hamiltonian structures.…”

Meta-Symplectic Geometry of 3rdOrder Monge-Ampère Equations and their Characteristics

“…Denomination rank-one refers to the rank of the multi-linear symmetric form on L involved in the definition (see, e.g., [13]). Observe that, in our contest, dim H α = 1, so that we shall speak of a characteristic line and call characteristic vector a generator of H α .…”

“…The primary motivation of this paper was to determine whether, and to what extent, such a phenomenon occurs also in the context of 3 rd order PDEs, where there can be found physically interesting analogues of the classical MAEs [1,13,14,15,28]. To the authors' best knowledge, such PDEs were defined by Boillat by using Lax's complete exceptionality [9,10,21].…”

On the quadratic normality and the triple curve of three-dimensional subvarieties of

“…A conjecture of Peskine and Van de Ven states that a smooth threefold X ⊂ P 5 is 2-normal unless it is the Palatini scroll (see, for instance, [10,Problem 5]). We remark that smoothness cannot be dropped according to [1]. On the other hand, the Palatini scroll is also the only known example of smooth threefold X in P 5 with reducible triple curve (cf.…”