Projective Duality, Unexpected Hypersurfaces and Logarithmic Derivations of Hyperplane Arrangements

Abstract

Several papers have been written studying unexpected hypersurfaces. We say a finite set of points Z admits unexpected hypersurfaces if a general union of fat linear subspaces imposes less that the expected number of conditions on the ideal of Z. In this paper, we introduce the concept of a very unexpected hypersurface. This is a stronger condition which takes into account an explanation for some hypersurfaces previously considered unexpected.We then develop a duality theory to relate the study of very unexpected hypersurfaces to the derivations of dual hyperplane arrangements. This allows us to generalize results in the plane of Cook, Harbourne, Migliore, Nagel [7] Faenzi, and Vallés [13] to higher dimensions. In particular, we give a criterion to determine if a set Z admits very unexpected hypersurfaces in the case Z is invariant under the action of an irreducible reflection group on the ambient projective space.Our approach has new applications even in P 2 where we are able to place strong conditions on sets of points Z which admit certain types of unexpected curves. We close relating Terao's Freeness Conjecture for line arrangements to a conjecture due to G. Dirac [9] on configurations of points in R 2 .

On $3$-syzygy and unexpected plane curves

“…The main purpose of the paper is to construct new examples of plane curves which are 3-syzygy curves, see Definition 5.7 -these are, for instance, nearly free curves, and use geometry standing behind them in order to show the existence of unexpected curves. Our results concerning unexpected curves provide a framework for constructing examples of such curves, and give additional insight into the results of Cook II-Harbourne-Migliore-Nagel [6] and of Trok [24].…”

Unexpected hypersurfaces and their consequences: A Survey

“…But it turns out that the new notion of unexpectedness has substantial intrinsic interest, with connections that touch a broad array of disciplines, including commutative algebra, combinatorics and representation theory, in addition to algebraic geometry. Here is some of the recent work on this topic [2,11,13,16,16,22,28,31,34,35,36,37].…”

“…The first two natural extensions of this idea are to move the problem to hypersurfaces in higher dimensional projective spaces, and to relax the restriction that the degree of the hypersurface should be one more than the multiplicity. As a first step, it has also been very productive to look at sets of points coming from other root systems, as was initiated in [22] (and continued in such work as [23,36]). However, the theory has gone far beyond root systems.…”

“…However, the theory has gone far beyond root systems. A third extension has been studied by Trok in [36]. He considers unexpected hypersurfaces that vanish on a general linear linear subspace with prescribed multiplicity and an arbitrary finite set of points.…”

“…A further extension of the concept of an unexpected hypersurface was developed by Trok in [36], building on [12,13,23]. Instead of considering a general point Trok studies the conditions imposed by a general linear subspace.…”