Projective Differential Geometry

Lane, Ernest P.

doi:10.1080/00029890.1933.11987494

Cited by 7 publications

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“…The theory of rationally connected varieties is quite recent and was formalized in [10], [36] and [35], although these varieties were intensively studied from different points of view by classical algebraic geometers, see for example [19], [59], [56], [6], [5], [42].…”

Section: Introductionmentioning

confidence: 99%

Varieties $n$-covered by curves of degree $\delta$

Pirio¹,

Russo²

2013

Comment. Math. Helv.

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Consequently, the maps I, G and T ω are projective automorphisms of the cubic X 3 J . Let Conf(J) be the conformal group of J, that is the subgroup of P GL(Z 2 (J)) generated by I and the maps G g and T ω for all ω ∈ J and all g ∈ Str(J). From Lemma 4.6, it follows that Conf(J) is a subgroup of the group of projective automorphisms of X 3 J .Proposition 4.7. The group of projective automorphisms of X 3 J acts transitively on 3uple's of general points of X 3 J . In fact, if x 1 , x 2 , x 3 ∈ J are sufficiently general, it exists µ ∈ Conf(J) such that µ(ν 3 (x 1 )) = 0 J , µ(ν 3 (x 2 )) = 1 J and µ(ν 3 (x 3 )) = ∞ J .Proof. First let G = Str(J) · e be the orbit in J of the unit e under the linear action of the structural group. By [61, Theorem 6.5], one knows that G is a Zariski-open subset of J.Assume that x, y, z ∈ J are such that (1) y 1 = y − x and z 1 = z − x are invertible; (2) z 2 = z −1 1 − y −1 1 ∈ G , i.e z 2 = g(e) for a certain g ∈ Str(J). Then setWe leave to the reader to verify that µ(ν 3 (x)) = ∞ J , µ(ν 3 (y)) = 0 J and µ(ν 3 (z)) = 1 J . Since ν 3 (J) is dense in X 3 J , the conclusion follows.Let x 1 , x 2 , x 3 ∈ J and let µ ∈ Conf(J) be as in the statement of Proposition 4.7. Since µ ∈ P GL(Z 2 (J)), the curve µ −1 (C J ) is a rational normal curve of degree 3 passing through the points ν 3 (x i ) for i = 1, 2, 3. Since µ is a projective automorphism of X 3 J (by Lemma 4.6 above), this twisted cubic curve is also contained in X 3 J . Thus we have proved the following result.Corollary 4.8. The twisted cubic X 3 J associated to a Jordan algebra J of rank three is 3-covered by rational normal curves of degree 3.Let k be the dimension of a Jordan algebra J of rank 3. Then X 3 J is a non-degenerate algebraic subvariety of the projective space PZ 2 (J), whose dimension is 2k + 1. Since π(k, 3, 3) = 2k + 2, one obtains that X 3 J ⊂ PZ 2 (J) = P 2k+1 is an example of X(k, 3, 3). Thus the cubics X 3 J associated to Jordan algebras of rank 3 are examples of varieties of

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Section: Introductionmentioning

confidence: 99%

Varieties $n$-covered by curves of degree $\delta$

Pirio¹,

Russo²

2013

Comment. Math. Helv.

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“…We refer to [33], [6], [14], [15], [13] for the further discussion of isothermally asymptotic surfaces.…”

Section: Isothermally Asymptotic Surfacesmentioning

confidence: 99%

“…For brevity, we only recall the necessary definitions. The details can be found in [6], [14], [33], etc. Thus, let us consider a point p 0 on the surface M 2 and the x-asymptotic line passing through p 0 .…”

Section: Projectively Minimal Surfacesmentioning

confidence: 99%

Integrable Systems in Projective Differential Geometry

Ferapontov

2000

Kyushu J. Math.

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Some of the most important classes of surfaces in projective 3-space are reviewed: these are isothermally asymptotic surfaces, projectively applicable surfaces, surfaces of Jonas, projectively minimal surfaces, etc. It is demonstrated that the corresponding projective "Gauss-Codazzi" equations reduce to integrable systems which are quite familiar from the modern soliton theory and coincide with the stationary flows in the Davey-Stewartson and Kadomtsev-Petviashvili hierarchies, equations of the Toda lattice, etc. The corresponding Lax pairs can be obtained by inserting a spectral parameter in the equations of the Wilczynski moving frame.

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“…Isothermally asymptotic surfaces arise as the focal surfaces of special W-congruences preserving Darboux's curves (zero curves of the Darboux cubic form) and are uniquely specified by the requirement that the 3-web formed by asymptotic and Darboux's curves is hexagonal. We refer to standard textbooks [5], [6], [7], [8], [10] for further discussion. Remark 1.…”

Section: Introductionmentioning

confidence: 99%

Stationary Veselov-Novikov equation and isothermally asymptotic surfaces in projective differential geometry

Ferapontov

1999

Differential Geometry and its Applications

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It is demonstrated that the stationary Veselov-Novikov (VN) and the stationary modified Veselov-Novikov (mVN) equations describe one and the same class of surfaces in projective differential geometry: the so-called isothermally asymptotic surfaces, examples of which include arbitrary quadrics and cubics, quartics of Kummer, projective transforms of affine spheres and rotation surfaces. The stationary mVN equation arises in the Wilczynski approach and plays the role of the projective "Gauss-Codazzi" equations, while the stationary VN equation follows from the Lelieuvre representation of surfaces in 3-space. This implies an explicit Bäcklund transformation between the stationary VN and mVN equations which is an analog of the Miura transformation between their (1+1)-dimensional limits.

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Projective Differential Geometry

Cited by 7 publications

References 0 publications

Varieties $n$-covered by curves of degree $\delta$

Varieties $n$-covered by curves of degree $\delta$

Integrable Systems in Projective Differential Geometry

Stationary Veselov-Novikov equation and isothermally asymptotic surfaces in projective differential geometry

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