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

Agafonov, Sergey I.; Ferapontov, E. V.

doi:10.1007/s229-001-8028-y

Cited by 18 publications

(71 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It may happen that the (n − 1)-dimensional submanifold M i degenerates into a linear subspace of codimension 2. This is closely related to the property for system (1) to possess Riemann invariants.…”

Section: -Component Systems Of Conservation Lawsmentioning

confidence: 84%

“…The classification of the corresponding T-systems is given in [1]. Systems of that type proved to be nondiagonalizable but integrable.…”

Section: Remark 2 General Linear Congruences In ‫ސ‬mentioning

confidence: 99%

“…It was observed recently that many constructions in the theory of systems of conservation laws (1) are, in a sense, parallel to that of the projective theory of congruences. The correspondence proposed in [3] and [4] associates with any system (1) an n-parameter family of lines…”

Section: Introduction Systems Of Conservation Laws Of Temple's Classmentioning

confidence: 99%

See 2 more Smart Citations

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

Agafonov¹,

Ferapontov²

2005

Glasgow Math. J.

View full text Add to dashboard Cite

Abstract. We propose a differential-geometric classification of the fourcomponent hyperbolic systems of conservation laws which satisfy the following properties: (a) they do not possess Riemann invariants; (b) they are linearly degenerate; (c) their rarefaction curves are rectilinear; (d) the cross-ratio of the four characteristic speeds is harmonic. This turns out to provide a classification of projective congruences in ‫ސ‬ 5 whose developable surfaces are planar pencils of lines, each of these lines cutting the focal variety at points forming a harmonic quadruplet. Symmetry properties and the connection of these congruences to Cartan's isoparametric hypersurfaces are discussed.2000 Mathematics Subject Classification. 53A25, 53B50, 35L65.

show abstract

Section: -Component Systems Of Conservation Lawsmentioning

confidence: 84%

“…The classification of the corresponding T-systems is given in [1]. Systems of that type proved to be nondiagonalizable but integrable.…”

Section: Remark 2 General Linear Congruences In ‫ސ‬mentioning

confidence: 99%

See 1 more Smart Citation

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

Agafonov¹,

Ferapontov²

2005

Glasgow Math. J.

View full text Add to dashboard Cite

show abstract

“…, a k ), k = [ n− 1 2 ], is called the multidegree of the congruence, the first integer a 0 is precisely its order. For instance, the multidegree of a linear congruence B in P 5 is (1,3,2), this means precisely that the lines of B contained in a general hyperplane fill a hypersurface of degree 3, and that the number of lines contained in a general P 3 is 2. Since B is formed precisely by the 4-secant lines of its focal locus X , a Palatini threefold in P 5 , we find in this way that X cannot be contained in a cubic hypersurface while, conversely, its hyperplane section is.…”

Section: Conjecturementioning

confidence: 99%

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

Poi

Mezzetti

2008

Geom Dedicata

View full text Add to dashboard Cite

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.

show abstract

“…These include the classification of varieties with one apparent double point ( [6,7]), the degree of irrationality of general hypersurfaces ( [4,5]) and in hyperbolic conservation laws, so called Temple systems of partial differential equations ( [3,10]). For a survey of order one congruences of lines, see [11].…”

Section: Introductionmentioning

confidence: 99%

Fano congruences of index 3 and alternating 3-forms

Poi¹,

Faenzi²,

Mezzetti³

et al. 2017

Ann. inst. Fourier

View full text Add to dashboard Cite

We study congruences of lines Xω defined by a sufficiently general choice of an alternating 3-form ω in n+1 dimensions, as Fano manifolds of index 3 and dimension n−1. These congruences include the G2-variety for n=6 and the variety of reductions of projected ℙ2×ℙ2 for n=7.\ud We compute the degree of Xω as the n-th Fine number and study the Hilbert scheme of these congruences proving that the choice of ω bijectively corresponds to Xω except when n=5. The fundamental locus of the congruence is also studied together with its singular locus: these varieties include the Coble cubic for n=8 and the Peskine variety for n=9.\ud The residual congruence Y of Xω with respect to a general linear congruence containing Xω is analysed in terms of the quadrics containing the linear span of Xω. We prove that Y is Cohen-Macaulay but non-Gorenstein in codimension 4. We also examine the fundamental locus G of Y of which we determine the singularities and the irreducible components

show abstract

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

Cited by 18 publications

References 21 publications

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

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

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

Fano congruences of index 3 and alternating 3-forms

Contact Info

Product

Resources

About