$(1,3,6,10,15,21,28,27,27,28)$ is a level $h$-vector! This example answers negatively the open question as to whether all codimension 3 level $h$-vectors are unimodal. Moreover, using the same (simple) technique, we are able to construct level algebras of codimension 3 whose $h$-vectors have exactly $N$ ` ` maxima", for any positive integer $N$. These non-unimodal $h$-vectors, in particular, provide examples of codimension 3 level algebras not enjoying the Weak Lefschetz Property (WLP). Their existence was also an open problem before. In the second part of the paper we further investigate this fundamental property, and show that there even exist codimension 3 level algebras of type 3 without the WLP.
The study of the $h$-vectors of graded Gorenstein algebras is an important topic in combinatorial commutative algebra, which despite the large amount of literature produced during the last several years, still presents many interesting open questions. In this note, we commence a study of those unimodal Gorenstein $h$-vectors that do \emph{not} satisfy the Stanley-Iarrobino property. Our main results, which are characteristic free, show that such $h$-vectors exist: 1) In socle degree $e$ if and only if $e\ge 6$; and 2) In every codimension five or greater. The main case that remains open is tha…
Improving the bounds of the Multiplicity Conjecture: the codimension 3 level case
Huneke和Srinivasan的多重性猜想（MC）给出了Cohen-Macaulay代数$A$的多重性的上、下界，它是由$A$的最小自由分解（MFR）模中出现的移位给出的.到目前为止研究的所有例子都导致了猜想（见$[HZ]$和$[MNR2]$），而且，MC的边界是尖锐的当且仅当$A$有一个纯MFR。因此，如果可能的话，寻找更好的（{\it ad hoc}）Cohen-Macaulay代数类的界似乎是一个合理且有用的想法。在这个工作中，我们将只考虑余维3的情况。在第一部分中，我们将坚持MC的界限，并证明它们对那些$h$-向量是压缩代数的$h$-向量的代数成立。在第二部分中，我们将（主要）集中在水平的情况下：我们将建立一个新的几何上的上限和下限的余维3水平代数$A$的重数，它可以表示为唯一的$A$的$h$-向量，这是优于（或等于）的MC提供。此外，即使当$A$的MFR不纯时，我们的边界也可以是尖锐的。即使证明我们的界限似乎仍然太困难的任务，在一般情况下，我们已经能够显示他们的一些有趣的类的余维3级代数$A$：即，当$A$被压缩，或当其$h$-向量$h(A)$结束与$(...,3,2)$。此外，我们将证明当$h(A)$以$(1,3,h_2,...)$开始时的下限，其中$h_2\leq 4$，以及当$h(A)$以$(...,h_{c-1},h_c)$结束时的上限，其中$h_{c-1}\…
Let $k$ be a field of characteristic $0$. Using the method of idealization, we show that there is a non-Koszul, quadratic, Artinian, Gorenstein, standard graded $k$-algebra of regularity $3$ and codimension $8$, answering a question of Mastroeni, Schenck, and Stillman. We also show that this example is minimal in the sense that no other idealization that is non-Koszul, quadratic, Artinian, Gorenstein algebra, with regularity $3$ has smaller codimension. We also construct an infinite family of graded, quadratic, Artinian, Gorenstein algebras $A_m$, indexed by an integer $m \ge 2$, with the foll…
We classify all possible $h$-vectors of graded artinian Gorenstein algebras in socle degree 4 and codimension $\leq 17$, and in socle degree 5 and codimension $\leq 25$. We obtain as a consequence that the least number of variables allowing the existence of a nonunimodal Gorenstein $h$-vector is 13 for socle degree 4, and 17 for socle degree 5. In particular, the smallest nonunimodal Gorenstein $h$-vector is $(1,13,12,13,1)$, which was constructed by Stanley in his 1978 seminal paper on level algebras. This solves a long-standing open question in this area. All of our results are characteristi…
We give examples of subcanonical subvarieties of codimension 3 in projective n-space which are not Pfaffian, i.e. defined by the ideal sheaf of submaximal Pfaffians of an alternating map of vector bundles. This gives a negative answer to a question asked by Okonek. Walter had previously shown that a very large majority of subcanonical subschemes of codimension 3 in P^n are Pfaffian, but he left open the question whether the exceptional non-Pfaffian cases actually occur. We give non-Pfaffian examples of the principal types allowed by his theorem, including (Enriques) surfaces in P^5 in characte…
