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arXiv · 2303.04665

Jacobian schemes of conic-line arrangements and eigenschemes

Valentina Beorchia·Rosa M. Miro'-Roig
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Abstract

The Jacobian scheme of a reduced, singular projective plane curve is the zero-dimensional scheme, whose homogeneous ideal is generated by the partials of its defining polynomial. The degree of such a scheme is called the global Tjurina number and, if the curve is not a set of concurrent lines, some upper and lower bounds depending on the degree of the curve and the minimal degree of a Jacobian syzygy, have been given by A.A. du Plessis and C.T.C. Wall. In this paper we give a complete geometric characterization of conic-line arrangenents, with global Tjurina number attaining the upper bound. Furthermore, we characterize conic-line arrangenents attaining the lower bound for the global Tjurina number, among all curves with a linear Jacobian syzygy. As an application, we characterize conic-line arrangenents with Jacobian scheme equal to an eigenscheme of some ternary tensor, and we study the geometry of their polar maps.

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Valentina Beorchia, Rosa M. Miro'-Roig. 2023-05-05. Jacobian schemes of conic-line arrangements and eigenschemes. https://arxiv.org/abs/2303.04665

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