Plane projections of a smooth space curve

• Autorzy: Trygve Johnsen
• 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 $a_1 ≥ 4$.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 1996
• Tom: 36
• Numer: 1
• Strony: 89-110

Daty

wydano

1996

Twórcy

autor

Trygve Johnsen

• Department of Mathematics, University of Bergen, Allégt. 55, N-5007 Bergen, Norway

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