AI 中文总结
该研究在$\boldsymbol{P}^3$中构造出含18个普通尖点的五次曲面，基于Barth–Rams分解，经有限域与特征零提升验证，为代数几何中奇异五次曲面的研究提供了新实例。
AI 中文摘要
我们在三维射影空间$\boldsymbol{P}^3$中构造具有18个普通尖点的五次曲面。研究起点是Barth–Rams对包含12个尖点的3-可除集的五次曲面的描述。对两个接触三次曲线沿两条异面直线奇异的情况进行特化，得到具有16个尖点的族，且在小有限域上可找到具有18个尖点的例子。我们的主要构造基于容许两种Barth–Rams分解的五次曲面，对应的12个尖点集交于7个点，且证明容许此类两种分解的五次曲面的轨迹在模空间中包含一个6维分支，其一般成员具有17个尖点。这使得在有限域上可高效找到具有18个尖点的成员。我们利用牛顿-亨塞尔提升和LLL重构将其中一个曲面提升到特征零，得到一个次数为22的数域上的五次曲面，并验证该曲面具有18个普通尖点且无其他奇点。
英文摘要
We construct quintic surfaces in the three-dimensional projective space $\mathbb P^3$ with $18$ ordinary cusps. Our starting point is the Barth--Rams description of quintics containing a $3$-divisible set of $12$ cusps. A specialization in which the two contact cubics are singular along two skew lines produces a family with $16$ cusps, and examples with $18$ cusps can be found over small finite fields. Our main construction is based on quintics admitting two Barth--Rams decompositions. The corresponding sets of $12$ cusps meet in $7$ points, and we prove that the locus of quintics admitting two such decompositions contains a $6$-dimensional component in the moduli space whose general member has $17$ cusps. This makes it possible to find members with $18$ cusps efficiently over finite fields. We lift one of these surfaces to characteristic zero using Newton--Hensel lifting and LLL reconstruction, obtaining a quintic over a number field of degree $22$. We verify that this surface has $18$ ordinary cusps and no other singularities.
