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Register a new userThe Hessian polynomial and the Jacobian ideal of a reduced hypersurface in $mathbb{P}^n$
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For a reduced hypersurface $V(f) subseteq mathbb{P}^n$ of degree $d$, the Castelnuovo-Mumford regularity of the Milnor algebra $M(f)$ is well understood when $V(f)$ is smooth, as well as when $V(f)$ has isolated singularities. We study the regularity of $M(f)$ when $V(f)$ has a positive dimensional singular locus. In certain situations, we prove that the regularity is bounded by $(d-2)(n+1)$, which is the degree of the Hessian polynomial of $f$. However, this is not always the case, and we prove that in $mathbb{P}^n$ the regularity of the Milnor algebra can grow quadratically in $d$.
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We reformulate a fundamental result due to Cook, Harbourne, Migliore and Nagel on the existence and irreduciblity of unexpected plane curves of a set of points $Z$ in $mathbb{P}^2$, using the minimal degree of a Jacobian syzygy of the defining equation for the dual line arrangement $mathcal A_Z$. Several applications of this new approach are given. In particular, we show that the irreducible unexpected quintics may occur only when the set $Z$ has the cardinality equal to 11 or 12, and describe five cases where this happens.
We describe the 0-th Fitting ideal of the Jacobian module of a plane curve in terms of determinants involving the Jacobian syzygies of this curve. This leads to new characterizations of maximal Tjurina curves, that is of non free plane curves, whose global Tjurina number equals an upper bound given by A. du Plessis and C.T.C. Wall.
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