# Plane projections of a smooth space curve

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

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## Abstract

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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 ${S}_{P}={\left({V}_{P}\right)}_{2}^{1}$ 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 ${a}_{1}\ge {a}_{2}\ge ...\ge {a}_{k}$ of $\Delta \left(d,g\right)=1/2\left(d-1\right)\left(d-2\right)-g$, 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 ${a}_{1}=\infty$. It is clear that the set of points P with ${a}_{1}\ge 2$ 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 ${a}_{1}\ge 3$. The second result describes the set of points with ${a}_{1}\ge 4$.

## How to cite

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Johnsen, 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

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8. [MS] T. Johnsen, Multiplicities of Solutions to some Enumerative Contact Problems, Math. Scand. 63 (1988), 87-108. Zbl0688.14051
9. [Pr] T. Johnsen, Plane projections of a smooth space curve, Preprint No. 8, Mathematics Reports, University of Tromsο, 1990.
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