Abstract
The computation of the dimension of linear systems of plane curves through a bunch of given multiple points is one of the most classic issues in algebraic geometry. In general, it is still an open problem to understand when the points fail to impose independent conditions. Despite many partial results, a complete solution is not known, even if the fixed points are in general position. The answer in the case of general points in the projective plane is predicted by the famous Segre–Harbourne–Gimigliano–Hirschowitz conjecture. When we consider fixed points in special position, even more interesting situations may occur. Recently, Di Gennaro, Ilardi and Vallès discovered a special configuration Z of nine points with a remarkable property: A general triple point always fails to impose independent conditions on the ideal of Z in degree four. The peculiar structure and properties of this kind of unexpected curves were studied by Cook II, Harbourne, Migliore and Nagel. By using both explicit geometric constructions and more abstract algebraic arguments, we classify low-degree unexpected curves. In particular, we prove that the aforementioned configuration Z is the unique one giving rise to an unexpected quartic.
Highlights
One of the central problems in algebraic geometry is the study of linear systems of hypersurfaces of Pn with imposed singularities, namely divisors containing a set of given points P1, . . . , Pr with multiplicities m1, . . . , mr
Let us start with the basic definitions in the projective plane
In [4], Cook II, Harbourne, Migliore and Nagel focused on a subtler problem about special linear systems of plane curves
Summary
One of the central problems in algebraic geometry is the study of linear systems of hypersurfaces of Pn with imposed singularities, namely divisors containing a set of given points P1, . . . , Pr with multiplicities m1, . . . , mr. In [4], Cook II, Harbourne, Migliore and Nagel focused on a subtler problem about special linear systems of plane curves. They drop the hypothesis of generality of some of the points, and they propose a classification problem analogous to the SHGH conjecture (see [4, Problem 1.4]), this problem seems too difficult to be solved in full generality. Things become more complicated for d = 4 In this case, there exists a configuration of nine points in P2 which admits an unexpected quartic. Theorem 1.6 Up to projective equivalence, the configuration of points Z ⊂ P2 in Example 1.5 is the only one which admits an unexpected curve of degree four. The stability of vector bundles turns out to be a powerful tool to prove Theorem 1.6
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