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Higher order Jacobians, Hessians and Milnor algebras. (English) Zbl 1448.13003
Let \(f \in R = \mathbb C[x_0,\dots,x_n]\) be a reduced form. Associated to \(f\) are a number of objects studied in this paper. The Jacobian ideal is \(J(f) = (\frac{\partial f}{\partial x_0},\dots,\frac{\partial f}{\partial x_n})\). The associated hypersurface is \(V(f) \subset \mathbb P^n\). The generators define the polar map \(\varphi_X : \mathbb P^n \dashrightarrow \mathbb P^n\), and \(Z_X := \varphi(\mathbb P^n)\) is the polar image of \(X\). The restriction \(\varphi_X | X = \mathcal G_X\) is the Gauss map of \(X\). The Milnor algebra of \(f\) is \(R/J(F)\). Via Macaulay-Matlis duality the annihilator of \(f\) is a homogeneous Gorenstein artinian algebra \(A(f)\) in the dual ring \(Q = \mathbb C[X_0,\dots,X_n]\). The Jacobian matrix associated to the polar map is the Hessian matrix of \(f\). In this paper the authors generalize these objects to higher order, such as the \(k\)-th order Milnor algebra \(M^k(f) := R/J^k\), and describe some geometry associated to them. They then study \(A(f)\) and \(M^k(f)\), for different choices of \(f\) and \(k\). In particular, they study the Hilbert function and Lefschetz properties of \(A(f)\) and \(M^k(f)\). They paint a nice picture of how these things all fit together.
13A02 Graded rings
05E40 Combinatorial aspects of commutative algebra
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13E10 Commutative Artinian rings and modules, finite-dimensional algebras
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