In the present note we study determinantal arrangements constructed with use of the 3-minors of a 3 × 5 generic matrix of indeterminates. In particular, we show that certain naturally constructed hypersurface arrangements in P 14 C are free.
In the note we study the multipoint Seshadri constants of O P 2 C (1) centered at singular loci of certain curve arrangements in the complex projective plane. Our first aim is to show that the values of Seshadri constants can be approximated with use of a combinatorial invariant which we call the configurational Seshadri constant. We study specific examples of point-curve configurations for which we provide actual values of the associated Seshadri constants. In particular, we provide an example based on Hesse point-conic configuration for which the associated Seshadri constant is computed by a line. This shows that multipoint Seshadri constants are not purely combinatorial.
The containment problem between symbolic and ordinary powers of homogeneous ideals has stimulated a lot of interesting research recently. In the most basic case of points in P 2 and powers I (3) and I 2 , there is a number of non-containment results based on arrangements of lines. In a joint paper with Lampa-Baczyńska we discovered the first example of non-containment based on an arrangement of axes and a singular irreducible curve of high degree.In the present note we show a similar example based on lines and a smooth curve of degree 6.2010 Mathematics Subject Classification. 14N20, 13A15, 13F20, 52C35 .
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