论文标题
在弱的lefschetz属性上，高度四个取代的完整交叉点
On the Weak Lefschetz Property for height four equigenerated complete intersections
论文作者
论文摘要
我们认为，所有Artinian Height的猜想是相同程度$ d $的形式的完整交叉点具有弱的Lefschetz属性（WLP）。我们将这个问题转化为一个研究$ \ Mathbb p^3 $的一般平滑曲线的一般超平面部分，我们的主要工具是Huneke和Ulrich的Socle Lemma以及仔细的联络参数。我们的主要结果是（i）证明该物业以$ d = 3,4 $和5; （ii）部分结果显示在非平凡但不完整的范围内最大等级，切成了先前未知范围的一半； （iii）证明最大等级在不同范围内的证明，即使没有假设所有发电机都具有相同的程度。我们还猜想，如果存在任何高度4 4个完整的交叉点，由相同程度和失败的WLP产生，那么必须存在一个（不一定是相同的）完全失败的一个（从某种意义上说，我们可以精确）。基于这个猜想，我们概述了一种证明四个变量中所有取代完整交集的WLP的方法。最后，我们将结果应用于〜$ \ Mathbb p^3 $的光滑表面的雅各布理想。
We consider the conjecture that all artinian height 4 complete intersections of forms of the same degree $d$ have the Weak Lefschetz Property (WLP). We translate this problem to one of studying the general hyperplane section of a certain smooth curve in $\mathbb P^3$, and our main tools are the Socle Lemma of Huneke and Ulrich together with a careful liaison argument. Our main results are (i) a proof that the property holds for $d=3,4$ and 5; (ii) a partial result showing maximal rank in a non-trivial but incomplete range, cutting in half the previous unknown range; and (iii) a proof that maximal rank holds in a different range, even without assuming that all the generators have the same degree. We furthermore conjecture that if there were to exist any height 4 complete intersection generated by forms of the same degree and failing WLP then there must exist one (not necessarily the same one) failing by exactly one (in a sense that we make precise). Based on this conjecture we outline an approach to proving WLP for all equigenerated complete intersections in four variables. Finally, we apply our results to the Jacobian ideal of a smooth surface in~$\mathbb P^3$.
