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The Jordan type of graded Artinian Gorenstein algebras. (English) Zbl 1441.13045

If \(A = R/I\) is a standard graded artinian algebra over a field \(K\), we say that \(A\) has the Weak Lefschetz Property (WLP) if there exists a linear form \(\ell\) for which \(\times \ell : [A]_i \rightarrow [A]_{i+1}\) has maximal rank for each \(i\). We say that it has the Strong Lefschetz Property (SLP) if there exists a linear form \(\ell\) for which \(\times \ell^d : [A]_i \rightarrow [A]_{i+d}\) has maximal rank for each \(i\) and each \(d \geq 1\). These properties have been extensively studied, especially in the case where \(A\) is Gorenstein, and many interesting open questions remain. The Jordan type of \(A\) is a related notion, from which WLP and SLP can be read. Let \(N = \dim_K A\). The Jordan type, \(\mathcal J_A\), is the partition of \(N\) given by the Jordan blocks of the multiplication \(\times \ell\) for a generic linear form \(\ell \in [A]_1\). This has also been studied extensively. In this paper the authors are concerned only with the case where \(A\) is Gorenstein. In this case it can be realized as the quotient of a ring of differential operators \(Q\) modulo the annihilator of a homogeneous form \(f\). The authors prove that the Jordan type of such an algebra depends only on the ranks of the mixed Hessians of \(f\), and give an algorithm to calculate the Jordan type of such an algebra. Finally, they use this algorithm and Gordan-Noether theory to classify all the possible Jordan types of artinian Gorenstein algebras of socle degree 3 or 4.

MSC:

13E10 Commutative Artinian rings and modules, finite-dimensional algebras
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
14M07 Low codimension problems in algebraic geometry
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References:

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