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本帖最后由 青青子衿 于 2023-3-17 08:15 编辑 射影平面上实有理三次曲线过八个点
Real rational cubics through 8 points in P2R
There are 12 singular (rational) cubic curves containing 8 general points in the plane.
Kontsevich’s famous recursion:
\begin{align*}
N_{d}&=\sum_{d_{1}+d_{2}=d}^{ }N_{d_1}N_{d_2}\left[d_{1}^2d_{2}^2\binom{3d-4}{3d_1-2}-d_{1}^{3}d_{2}\binom{3d-4}{3d_1-1}\right]\\
N_3&=\sum_{(d_1,d_2)=(1,2),(2,1)}^{ }N_{d_1}N_{d_2}\left[d_{1}^2d_{2}^2\binom{3d-4}{3d_1-2}-d_{1}^{3}d_{2}\binom{3d-4}{3d_1-1}\right]\\
&=N_{1}N_{2}\left[1\cdot4\cdot\binom{5}{1}-1^{3}\cdot2\cdot\binom{5}{2}\right]+N_{2}N_{1}\left[4\cdot1\cdot\binom{5}{4}-2^{3}\cdot1\cdot\binom{5}{5}\right]\\
&=20-20+20-8=12
\end{align*}
OEIS A013587
Number of rational plane curves of degree d passing through 3d-1 general points.
oeis.org/A013587
枚举实代数几何
Enumerative Real Algebraic Geometry 3.iv.
math.tamu.edu/~sottile/research/pages/ERAG/S3/4.html |
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