论文标题
枚举几何形状
Equivariant enumerative geometry
论文作者
论文摘要
我们制定了数字的模棱两可的守恒，这证明可以将复杂的载体矢量捆绑包的广义欧拉数量计算为任意部分的局部指数之和。这涉及在模棱两可的环境中扩展Pontryagin- Thom转移。我们利用这一结果开始研究在群体作用的情况下进行枚举几何形状的研究。为了说明这种机械的力量，我们证明，由对称多项式定义的任何光滑的复杂立方表面都有27行，其轨道类型下的轨道类型$ \ s_4 $ - $ \ mathbb {c} p^3 $给出了$ [s_4/c_2] + [s_4/c_2] + [s_4/c_2' + [s_4/c_2'$ [$ [s_4/d $ d_4/d_2 $ de_4/d $ d_8] $ c。二级的非偶联循环子组。结果，我们证明了真实的对称立方表面只能包含3或27个真实线。
We formulate an equivariant conservation of number, which proves that a generalized Euler number of a complex equivariant vector bundle can be computed as a sum of local indices of an arbitrary section. This involves an expansion of the Pontryagin--Thom transfer in the equivariant setting. We leverage this result to commence a study of enumerative geometry in the presence of a group action. As an illustration of the power of this machinery, we prove that any smooth complex cubic surface defined by a symmetric polynomial has 27 lines whose orbit types under the $S_4$-action on $\mathbb{C}P^3$ are given by $[S_4/C_2]+[S_4/C_2'] + [S_4/D_8]$, where $C_2$ and $C_2'$ denote two non-conjugate cyclic subgroups of order two. As a consequence we demonstrate that a real symmetric cubic surface can only contain 3 or 27 real lines.
