Strange curves

Bayer, Valmecir Antonio dos Santos; Hefez, Abramo

doi:10.1080/00927879108824305

Cited by 21 publications

(29 citation statements)

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“…Strangeness is a phenomenon in characteristic p > 0. For the geometry of strange plane curves, see Bayer-Hefez [ 1 ] and Hefez-Vainsencher [5]. We will consider a condition on a plane curve which is weaker than strangeness.…”

Section: Introductionmentioning

confidence: 99%

Plane curves whose tangent lines at collinear points are concurrent

Homma

1994

Geom Dedicata

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Abstract.We are interested in a particular geometry of plane curves in characteristic p > 0, which was inspired by Thas's article [13]. We will prove that any plane curve of degree > 2 whose tangent lines at collinear points are concurrent is either a strange curve or projectively equivalent to the Fermat curve of degree q + 1, where q is a power ofp.Mathematics Subject Classifications (1991): Primary 14N05; secondary 51N05, 14H99.

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Section: Introductionmentioning

confidence: 99%

Plane curves whose tangent lines at collinear points are concurrent

Homma

1994

Geom Dedicata

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“…Such curves were studied by E. Lluis in [9], showing that all strange curves are singular, except for conics in characteristic two, see also [11] and [6], [2], [1], for further developments.…”

Section: Introductionmentioning

confidence: 99%

“…Let X ⊂ P n , n м 3, be an integral non-degenerate projective curve and let P a regular point of X. Let T (2) P (X ) be the osculating plane to X at P.…”

Section: Introductionmentioning

confidence: 99%

“…In characteristic zero, the osculating plane to a generic point P ∈ X is such that length(X ∩ T (2) P (X)) = 3, i.e., a general osculating plane is not hyperosculating. However, in positive characteristic, there are non-degenerate curves X ⊂ P n such that, for a general P ∈ X, length(X ∩ T (2) P (X)) м 5 (see [8]).…”

Section: Introductionmentioning

confidence: 99%

“…However, in positive characteristic, there are non-degenerate curves X ⊂ P n such that, for a general P ∈ X, length(X ∩ T (2) P (X)) м 5 (see [8]). Notice that the generic osculating conic to X (which exists if b 2 м 5) is the double line of T (2) P (X) if and only if b 1 > 2 and it is smooth if b 1 = 2.…”

Section: Introductionmentioning

confidence: 99%

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Two-strange projective curves

Ballico¹,

Cossidente²

2002

Archiv der Mathematik

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The notion of second order strange projective curve in P n , n м 3, is introduced. A non-degenerate curve C in P n , n м 3, is said to be a second order strange curve if it admits osculating conics and such that all its osculating conics pass through a common point called 2-strange point. An existence-construction result is provided and some examples are furnished. Introduction.Let K be an algebraically closed field of characteristic p > 0. A strange curve in the n-dimensional projective space P n is a non-linear irreducible projective curve, with the property that all its tangent lines at simple points pass through a given point called center or strange point.Such curves were studied by E. Lluis in [9], showing that all strange curves are singular, except for conics in characteristic two, see also [11] and [6], [2], [1], for further developments.In this paper, we present a generalization of strange curves. Precisely, we introduce the notion of second order strange curve in P n .A non-degenerate curve C in P n , n м 3, is said to be a second order strange curve (or 2-strange curve) if it admits osculating conics and such that all its osculating conics pass through a common point (2-strange point).We stress that this is a characteristic p phenomenon not possible in characteristic zero and as far as we know, this is the stronger and more general non-linear generalization of the notion of strange curve. We are able not only to prove the existence of such curves, but also to construct them explicitly. Our curves are contained in cones of P n with vertex a point P and basis a curve in a hyperplane H of P n , not containing P. Choosing any curve D with a birational model in H, we find a curve C in the cone over D which is birational to D.We recall some preliminary notions. Let C be a non-degenerate irreducible curve in P n . Then for a general point P of C, there is a complete flag of linear subspacesis called the (Hasse) sequence of invariants of C , [12], [8], [14], [7], [3].In Section 2 we discuss the notion of generic osculating conic to a curve at a point P, giving conditions on uniqueness.

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Very Strange Projective Curves

Ballico

2001

Results. Math.

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Strange curves

Cited by 21 publications

References 3 publications

Plane curves whose tangent lines at collinear points are concurrent

Plane curves whose tangent lines at collinear points are concurrent

Two-strange projective curves

Very Strange Projective Curves

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