# Plane projections of a smooth space curve

Banach Center Publications (1996)

- Volume: 36, Issue: 1, page 89-110
- ISSN: 0137-6934

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topJohnsen, Trygve. "Plane projections of a smooth space curve." Banach Center Publications 36.1 (1996): 89-110. <http://eudml.org/doc/208586>.

@article{Johnsen1996,

abstract = {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 $a_1 ≥ 4$.},

author = {Johnsen, Trygve},

journal = {Banach Center Publications},

keywords = {equisingularity; space curve},

language = {eng},

number = {1},

pages = {89-110},

title = {Plane projections of a smooth space curve},

url = {http://eudml.org/doc/208586},

volume = {36},

year = {1996},

}

TY - JOUR

AU - Johnsen, Trygve

TI - Plane projections of a smooth space curve

JO - Banach Center Publications

PY - 1996

VL - 36

IS - 1

SP - 89

EP - 110

AB - 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 $a_1 ≥ 4$.

LA - eng

KW - equisingularity; space curve

UR - http://eudml.org/doc/208586

ER -

## References

top- [ACGH] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves, Vol. I. Springer Verlag, New York, 1985. Zbl0559.14017
- [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. Zbl0494.51016
- [GH] P. Griffiths, J. D. Harris, Principles of Algebraic Geometry, Pure & Applied Mathematics, Wiley, 1978. Zbl0408.14001
- [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. Zbl0701.14045
- [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. Zbl0688.14051
- [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. Zbl0314.14003
- [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.

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