斐波那契数与平面分支的self-dual格结构
Fibonacci numbers and self-dual lattice structures for plane branches
摘要：
考虑光滑复解析曲面上的平面分支，即曲线的不可约芽。我们将其爆破复杂度定义为实现其最小嵌入分辨率所需的点的爆破数量。我们证明了爆破复杂性$n$存在$F_{2n-4}$拓扑型，其中$F_{n}$是$n$的Fibonacci数.我们引入复杂性保持操作的拓扑型，增加的多重性，我们推导出的最大多重性的平面分支的爆破复杂性$n$是$F_n$。它是通过两种拓扑型来实现的，其中一种被认为是唯一能使米尔诺尔数最大化的拓扑型。我们还证明了在爆破复杂性$n$的平面分支的拓扑型集合上存在一个自然的偏序关系，使这个集合成为一个分配格，即它的任何两个元素都有一个下确界和一个上确界，这些运算中的每一个相对于第二个都是分配的.我们证明了这个格允许一个唯一的反序双射。由于这个双射是对合的，它定义了平面分支的拓扑型的对偶。最大化Milnor数的类型也是这个格的极大元，它的对偶是唯一具有最小Milnor数的类型。存在爆破复杂性$n$的$F_{n-2}$自对偶拓扑型。我们的证明是通过编码的拓扑型相关联的恩里克图。
英文摘要：
Consider a plane branch, that is, an irreducible germ of curve on a smooth complex analytic surface. We define its blow-up complexity as the number of blow-ups of points necessary to achieve its minimal embedded resolution. We show that there are $F_{2n-4}$ topological types of blow-up complexity $n$, where $F_{n}$ is the $n$-th Fibonacci number. We introduce complexity-preserving operations on topological types which increase the multiplicity and we deduce that the maximal multiplicity for a plane branch of blow-up complexity $n$ is $F_n$. It is achieved by exactly two topological types, one of them being distinguished as the only type which maximizes the Milnor number. We show moreover that there exists a natural partial order relation on the set of topological types of plane branches of blow-up complexity $n$, making this set a distributive lattice, that is, any two of its elements admit an infimum and a supremum, each one of these operations beeing distributive relative to the second one. We prove that this lattice admits a unique order-inverting bijection. As this bijection is involutive, it defines a duality for topological types of plane branches. The type which maximizes the Milnor number is also the maximal element of this lattice and its dual is the unique type with minimal Milnor number. There are $F_{n-2}$ self-dual topological types of blow-up complexity $n$. Our proofs are done by encoding the topological types by the associated Enriques diagrams.
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