11. Über die Wendepuncte der Curven dritter Ordnung.

Hesse, O.

doi:10.1515/9783112368268-011

Cited by 11 publications

(29 citation statements)

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“…It is well-known ( [3], [4]) that the set of nine inflection points of a plane cubic curve is a projectively rigid set, that is, for each two smooth plane cubic curves C 1 and C 2 there is a projective transformation of the plane sending the set of the inflection points of C 1 onto the set of the inflection points of C 2 . Therefore there is an imbedding ϕ : P GL(3, C) → (P 2 ) 9 given for τ ∈ P GL(3, C) by ϕ : τ → (τ (q 1 ), .…”

Section: Proof Consider the Family Of Curvesmentioning

confidence: 99%

On the variety of the inflection points of plane cubic curves

Kulikov¹

2019

Izv. Math.

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In the paper, we investigate properties of the nine-dimensional variety of the inflection points of the plane cubic curves. The description of local monodromy groups of the set of inflection points near singular cubic curves is given. Also, it is given a detailed description of the normalizations of the surfaces of the inflection points of plane cubic curves belonging to general two-dimensional linear systems of cubic curves, The vanishing of the irregularity a smooth manifold birationally isomorphic to the variety of the inflection points of the plane cubic curves is proved.

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Section: Proof Consider the Family Of Curvesmentioning

confidence: 99%

On the variety of the inflection points of plane cubic curves

Kulikov¹

2019

Izv. Math.

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“…Each curve composing the pencil is called the Hesse cubic curve and is denoted by E t 0 ,t 1 . It is well known that every nonsingular cubic curve can be transformed projectively into a member of the Hesse pencil {E t 0 ,t 1 } (t 0 ,t 1 )∈P 1 (C) [4,5].…”

Section: Hesse Pencilmentioning

confidence: 99%

“…The correspondence (4), (5), and ( 6) in terms of η tell us that the addition with p 6 corresponds to that with V 3 on Cκ , while that with p 1 vanishes in the limit ε → 0. (Note that V 1 is the unit of addition on Cκ .)…”

Section: Tropicalization Of the Hessian Groupmentioning

confidence: 99%

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Group actions on the tropical Hesse pencil

Nobe

2016

Japan J. Indust. Appl. Math.

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We show that the addition of points on the tropical Hesse curve can be realized via the intersection with a tropical line. Then the addition formula for the tropical Hesse curve is reduced from those for the level-three theta functions through the ultradiscretization procedure. A tropical analogue of the Hessian group G 216 , the group of linear automorphisms acting on the Hesse pencil, is also investigated; it is shown that the dihedral group D 3 of degree three is the group of linear automorphisms acting on the tropical Hesse pencil.

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“…Hesse [20], and before him Plücker [21], studied this family of curves in detail. Their first object was to determine the inflection points.…”

Section: The Syzygetic Hesse Pencilmentioning

confidence: 99%

From SICs and MUBs to Eddington

Bengtsson

2010

J. Phys.: Conf. Ser.

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This is a survey of some very old knowledge about Mutually Unbiased Bases (MUB) and Symmetric Informationally Complete POVMs (SIC). In prime dimensions the former are closely tied to an elliptic normal curve symmetric under the Heisenberg group, while the latter are believed to be orbits under the Heisenberg group in all dimensions. In dimensions 3 and 4 the SICs are understandable in terms of elliptic curves, but a general statement escapes us. The geometry of the SICs in 3 and 4 dimensions is discussed in some detail.

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11. Über die Wendepuncte der Curven dritter Ordnung.

Cited by 11 publications

References 0 publications

On the variety of the inflection points of plane cubic curves

On the variety of the inflection points of plane cubic curves

Group actions on the tropical Hesse pencil

From SICs and MUBs to Eddington

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