Illinois Journal of Mathematics

On the number of real hypersurfaces hypertangent to a given real space curve

J. Huisman

Full-text: Open access

Abstract

Let $C$ be a smooth geometrically integral real algebraic curve in projective $n$-space $\mathbb{P}^n$. Let $c$ be its degree and let $g$ be its genus. Let $d$, $s$ and $m$ be nonzero natural integers. Let $\nu$ be the number of real hypersurfaces of degree $d$ that are tangent to at least $s$ real branches of $C$ with order of tangency at least $m$. We show that $\nu$ is finite if $s=g$, $gm=cd$ and the restriction map $H^0(\mathbb{P}^n,\mathcal{O}(d))\rightarrow H^0(C,\mathcal{O}(d))$ is an isomorphism. Moreover, we determine explicitly the value of $\nu$ in that case.

Article information

Source
Illinois J. Math., Volume 46, Number 1 (2002), 145-153.

Dates
First available in Project Euclid: 13 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258136145

Digital Object Identifier
doi:10.1215/ijm/1258136145

Mathematical Reviews number (MathSciNet)
MR1936080

Zentralblatt MATH identifier
1007.14011

Subjects
Primary: 14P05: Real algebraic sets [See also 12D15, 13J30]
Secondary: 14N10: Enumerative problems (combinatorial problems)

Citation

Huisman, J. On the number of real hypersurfaces hypertangent to a given real space curve. Illinois J. Math. 46 (2002), no. 1, 145--153. doi:10.1215/ijm/1258136145. https://projecteuclid.org/euclid.ijm/1258136145


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