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2004/12/02 | 常二重点[Ordinary Double Point]
类别(∑〖数学〗)
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发表于 13:13
Let
(or
) be a
space curve
. Then a point
(where
denotes the
immersion
of
f
) is an ordinary double point if its
preimage
under
f
consists of two values
and
, and the two
tangent vectors
and
are noncollinear. Geometrically, this means that, in a
neighborhood
of
p
, the curve consists of two transverse branches. Ordinary double points are
isolated singularities
having
Coxeter-Dynkin diagram
of type
, and also called "nodes" or "simple double points."
The above plot shows the curve
, which has an ordinary double point at the
origin
.
A surface in complex three-space admits at most finitely many ordinary double points. The maximum possible number of ordinary double points
for a surface of degree
d
= 1,2, ..., are 0, 1, 4, 16, 31, 65,
,
,
,
,
,
... (Sloane's
A046001
;Chmutov 1992, Endraß 1995).
was known to Kummer in 1864 (Chmutov 1992), the fact that
was proved by Beauville (1980), and
was proved by Jaffe and Ruberman (1994). For
, the following inequality holds:
(Endraß 1995). Examples of
algebraic surfaces
having the maximum (known) number of ordinary double points are given in the following table.
<DIV ALIGN="CENTER"> <TABLE CELLPADDING=3 BORDER="1"> <TR><TD ALIGN="RIGHT"><i>d</i></TD> <TD ALIGN="RIGHT"><IMG WIDTH="34" HEIGHT="31" ALIGN="MIDDLE" BORDER="0" SRC="http://mathworld.wolfram.com/oimg1154.gif"></TD> <TD ALIGN="LEFT">Surface</TD> </TR> <TR><TD ALIGN="RIGHT">3</TD> <TD ALIGN="RIGHT">4</TD> <TD ALIGN="LEFT"><a href="CayleyCubic.html">Cayley cubic</a></TD> </TR> <TR><TD ALIGN="RIGHT">4</TD> <TD ALIGN="RIGHT">16</TD> <TD ALIGN="LEFT"><a href="KummerSurface.html">Kummer surface</a></TD> </TR> <TR><TD ALIGN="RIGHT">5</TD> <TD ALIGN="RIGHT">31</TD> <TD ALIGN="LEFT"><a href="Dervish.html">dervish</a></TD> </TR> <TR><TD ALIGN="RIGHT">6</TD> <TD ALIGN="RIGHT">65</TD> <TD ALIGN="LEFT"><a href="BarthSextic.html">Barth sextic</a></TD> </TR> <TR><TD ALIGN="RIGHT">7</TD> <TD ALIGN="RIGHT">93</TD> <TD ALIGN="LEFT"><a href="ChmutovSurface.html">Chmutov surface</a></TD> </TR> <TR><TD ALIGN="RIGHT">8</TD> <TD ALIGN="RIGHT">168</TD> <TD ALIGN="LEFT"><a href="EndrassOctic.html">Endraß octic</a></TD> </TR> <TR><TD ALIGN="RIGHT">9</TD> <TD ALIGN="RIGHT">216</TD> <TD ALIGN="LEFT"><a href="ChmutovSurface.html">Chmutov surface</a></TD> </TR> <TR><TD ALIGN="RIGHT">10</TD> <TD ALIGN="RIGHT">345</TD> <TD ALIGN="LEFT"><a href="BarthDecic.html">Barth decic</a></TD> </TR> <TR><TD ALIGN="RIGHT">11</TD> <TD ALIGN="RIGHT">425</TD> <TD ALIGN="LEFT"><a href="ChmutovSurface.html">Chmutov surface</a></TD> </TR> <TR><TD ALIGN="RIGHT">12</TD> <TD ALIGN="RIGHT">600</TD> <TD ALIGN="LEFT"><a href="SartiDodecic.html">Sarti dodecic</a></TD> </TR> </TABLE> </DIV>
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