Plane projections of a smooth space curve

Johnsen, Trygve

Johnsen, Trygve

Banach Center Publications, Tome 37 (1996), p. 89-110 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $S_{P} = {(V_{P})}_{2}^{1}$ 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 $a_{1} \geq a_{2} \geq . . . \geq a_{k}$ 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 $a_{1} = \infty$ . It is clear that the set of points P with $a_{1} \geq 2$ 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 $a_{1} \geq 3$ . The second result describes the set of points with $a_{1} \geq 4$ .

Publié le : 1996-01-01

Zbl 0876.14020

EUDML-ID : urn:eudml:doc:208586

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     author = {Johnsen, Trygve},
     title = {Plane projections of a smooth space curve},
     journal = {Banach Center Publications},
     volume = {37},
     year = {1996},
     pages = {89-110},
     zbl = {0876.14020},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv36z1p89bwm}
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Johnsen, Trygve. Plane projections of a smooth space curve. Banach Center Publications, Tome 37 (1996) pp. 89-110. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv36z1p89bwm/

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