关于$k[x,y,z]$中一般线性型一致幂的强Lefschetz问题
On the strong Lefschetz question for uniform powers of general linear forms in $k[x,y,z]$
摘要：
Schenck和Seceleanu证明了，如果$R = k[x,y,z]$，其中$k$是一个无限域，$I$是一个由任何线性型的幂的集合生成的理想，那么乘以一般线性型$L$会导致从$R/I$的任何分量到下一个分量的最大秩同态。也就是说，$R/I$具有{ \em弱Lefschetz性质}。考虑到更一般的{ \em强Lefschetz问题}，当$\times L^j$对$j \geq 2$有最大秩时，我们首次对这个问题进行了系统的研究.我们假设线性型是一般的，并且幂都是相同的，即$I$是由一般线性型的{PDMATHAMR uniform}幂生成的。我们证明了，对于任何数量的这样的生成器，$\times L^2$总是有最大的秩。然后，我们专门几乎完全的交叉点，即四个发电机，我们表明，$j = 3,4,5$的行为取决于统一的指数和$j$，在某种程度上，我们使精确。特别地，总是存在$\times L^j$不符合最大秩的最多一个度。最后，我们注意到，实验上所有较高的权力$L$失败最大排名在至少两个学位。
英文摘要：
Schenck and Seceleanu proved that if $R = k[x,y,z]$, where $k$ is an infinite field, and $I$ is an ideal generated by any collection of powers of linear forms, then multiplication by a general linear form $L$ induces a homomorphism of maximal rank from any component of $R/I$ to the next. That is, $R/I$ has the {\em weak Lefschetz property}. Considering the more general {\em strong Lefschetz question} of when $\times L^j$ has maximal rank for $j \geq 2$, we give the first systematic study of this problem. We assume that the linear forms are general and that the powers are all the same, i.e. that $I$ is generated by {\em uniform} powers of general linear forms. We prove that for any number of such generators, $\times L^2$ always has maximal rank. We then specialize to almost complete intersections, i.e. to four generators, and we show that for $j = 3,4,5$ the behavior depends on the uniform exponent and on $j$, in a way that we make precise. In particular, there is always at most one degree where $\times L^j$ fails maximal rank. Finally, we note that experimentally all higher powers of $L$ fail maximal rank in at least two degrees.
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