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====Comments====
 
====Comments====
In algebraic geometry, a unicursal curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095170/u09517013.png" /> is a rational curve, i.e. a curve that admits a parametric representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095170/u09517014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095170/u09517015.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095170/u09517016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095170/u09517017.png" /> rational functions. Such a curve is an algebraic curve of effective genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095170/u09517018.png" />. For every irreducible curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095170/u09517019.png" /> there exists a birationally equivalent non-singular curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095170/u09517020.png" />. This <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095170/u09517021.png" /> is unique up to isomorphism. The genus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095170/u09517022.png" /> is called the effective genus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095170/u09517023.png" />. The unicursal curves are the irreducible algebraic curves of effective genus zero. This (more or less) agrees with the general geometric definition above, in that the parametrization provides a "traversion" .
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In algebraic geometry, a unicursal curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095170/u09517013.png" /> is a rational curve, i.e. a curve that admits a parametric representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095170/u09517014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095170/u09517015.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095170/u09517016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095170/u09517017.png" /> rational functions. Such a curve is an algebraic curve of effective genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095170/u09517018.png" />. For every irreducible curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095170/u09517019.png" /> there exists a birationally equivalent non-singular curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095170/u09517020.png" />. This <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095170/u09517021.png" /> is unique up to isomorphism. The genus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095170/u09517022.png" /> is called the effective genus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095170/u09517023.png" />. The unicursal curves are the irreducible algebraic curves of effective genus zero. This (more or less) agrees with the general geometric definition above, in that the parametrization provides a "traversion" .
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.J. Walker,   "Algebraic curves" , Dover, reprint (1950) pp. 149–151</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P.A. Griffiths,   J.E. Harris,   "Principles of algebraic geometry" , Wiley (Interscience) (1978) pp. 178; 674; 179; 349; 525; 532; 535; 632; 743</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.J. Walker, "Algebraic curves" , Dover, reprint (1950) pp. 149–151 {{MR|0033083}} {{ZBL|0039.37701}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) pp. 178; 674; 179; 349; 525; 532; 535; 632; 743 {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR></table>

Revision as of 21:57, 30 March 2012

A plane curve which may be traversed such that the points of self-intersection are visited only twice. For a curve to be unicursal it is necessary and sufficient that there are at most two points through which there pass an odd number of paths. If is a plane algebraic curve of order having the maximum number of double points (including improper and imaginary ones), then (where a point of multiplicity is counted as double points).

Every integral , where is the function of defined by the equation giving an algebraic unicursal curve and is a rational function, can be reduced to an integral of a rational function and can be expressed in terms of elementary functions.


Comments

In algebraic geometry, a unicursal curve is a rational curve, i.e. a curve that admits a parametric representation , with and rational functions. Such a curve is an algebraic curve of effective genus . For every irreducible curve there exists a birationally equivalent non-singular curve . This is unique up to isomorphism. The genus of is called the effective genus of . The unicursal curves are the irreducible algebraic curves of effective genus zero. This (more or less) agrees with the general geometric definition above, in that the parametrization provides a "traversion" .

References

[a1] R.J. Walker, "Algebraic curves" , Dover, reprint (1950) pp. 149–151 MR0033083 Zbl 0039.37701
[a2] P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) pp. 178; 674; 179; 349; 525; 532; 535; 632; 743 MR0507725 Zbl 0408.14001
How to Cite This Entry:
Unicursal curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unicursal_curve&oldid=24002
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article