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## Banach Center Publications

1996 | 36 | 1 | 89-110
Tytuł artykułu

### Plane projections of a smooth space curve

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let C be a smooth non-degenerate integral curve of degree d and genus g in $ℙ^3$ over an algebraically closed field of characteristic zero. For each point P in $ℙ^3$ let $V_P$ be the linear system on C induced by the hyperplanes through P. By $V_P$ one maps C onto a plane curve $C_P$, such a map can be seen as a projection of C from P. If P is not the vertex of a cone of bisecant lines, then $C_P$ will have only finitely many singular points; or to put it slightly different: The secant scheme $S_P = (V_P)^1_2$ parametrizing divisors in the second symmetric product $C_2$ that fail to impose independent conditions on $V_P$ will be finite. Hence each such point P gives rise to a partition ${a_1 ≥ a_2 ≥ ... ≥ a_k}$ of $Δ(d,g) = 1/2(d-1)(d-2)-g$, where the $a_i$ are the local multiplicities of the scheme $S_P$. If P is the vertex of a cone of bisecant lines (for example if P is a point of C), we set $a_1 = ∞$. It is clear that the set of points P with $a_1 ≥ 2$ is the surface F of stationary bisecant lines (including some tangent lines); a generic point P on F gives a tacnodial $C_P$. We give two results valid for all curves C. The first one describes the set of points P with $a_1 ≥ 3$. The second result describes the set of points with $a_1 ≥ 4$.
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Rocznik
Tom
Numer
Strony
89-110
Opis fizyczny
Daty
wydano
1996
Twórcy
autor
• Department of Mathematics, University of Bergen, Allégt. 55, N-5007 Bergen, Norway
Bibliografia
• [ACGH] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves, Vol. I. Springer Verlag, New York, 1985.
• [BM] W. Barth, R. Moore, On rational plane sextics with six tritangents, in: Algebraic Geometry and Commutative Algebra, Vol. I, 45-58, Kinokuniya, Tokyo, 1988.
• [Bo] J. M. Boardman, Singularities of Differentiable Maps, Inst. Hautes Études Sci. Publ. Math. 33 (1967), 21-57.
• [vzG] J. von zur Gathen, Secant Spaces to Curves, Canad. J. Math. 35 (1983), 589-612.
• [GH] P. Griffiths, J. D. Harris, Principles of Algebraic Geometry, Pure & Applied Mathematics, Wiley, 1978.
• [LP-I] M. E. Huibregtse, T. Johnsen, Local Properties of Secant Varieties in Symmetric Products, Part I. Trans. Amer. Math. Soc. 313 (1989), 187-204.
• [LP-II] T. Johnsen, Local Properties of Secant Varieties in Symmetric Products, Part II. Trans. Amer. Math. Soc. 313 (1989), 205-220.
• [MS] T. Johnsen, Multiplicities of Solutions to some Enumerative Contact Problems, Math. Scand. 63 (1988), 87-108.
• [Pr] T. Johnsen, Plane projections of a smooth space curve, Preprint No. 8, Mathematics Reports, University of Tromsο, 1990.
• [Ro] J. Roberts, Singularity Subschemes and Generic Projections, Trans. Amer. Math. Soc. 212 (1975), 229-268.
• [Te] B. Teissier, The Hunting of Invariants in the Geometry of Discriminants, Proc. Nordic Summer School; Real and Complex Singularities, Oslo 1976, Ed. Per Holm, Sijthoff and Hoordhoff, 1977.
• [Th] R. Thom, Les Singularités des Applications Différentiables, Ann. Inst. Fourier (Grenoble) 6 (1955/56), 43-87.
Typ dokumentu
Bibliografia
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