Plane projections of a smooth space curve

@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 -