Unexpected surfaces singular on lines in $${\mathbb {P}}^{3}$$ P 3
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Unexpected surfaces singular on lines in P3 Marcin Dumnicki1 · Brian Harbourne2 · Joaquim Roé3 · Tomasz Szemberg4 1 Halszka Tutaj-Gasinska ´
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Received: 15 December 2019 / Revised: 25 July 2020 / Accepted: 25 August 2020 © The Author(s) 2020
Abstract We study linear systems of surfaces in P3 singular along general lines. Our purpose is to identify and classify special systems of such surfaces, i.e., those non-empty systems where the conditions imposed by the multiple lines are not independent. We prove the existence of four surfaces arising as (projective) linear systems with a single reduced member. Till now no such examples have been known. These are unexpected surfaces in the sense of recent work of Cook II, Harbourne, Migliore, and Nagel. It is an open problem if our list is complete, i.e., if it contains all reduced and irreducible unexpected surfaces based on lines in P3 . As an application we find Waldschmidt constants of six general lines in P3 and an upper bound for this invariant for seven general lines. Keywords Fat flats · Special linear systems · Unexpected varieties · Base loci · Cremona transformations Mathematics Subject Classification 14C20 · 14E05
1 Introduction The study of linear systems of hypersurfaces in complex projective spaces with assigned base points of given multiplicity is a classical and central problem in algebraic geometry; see, e.g. [2,12,21,28]. In the last few years this problem has been generalized to linear systems of hypersurfaces with assigned base loci consisting of linear subspaces of higher dimension [16,19]. Conjectures such as [19, Conjecture 5.5] and [16, Conjectures A, B and C] suggest that the asymptotic behavior of such linear systems in the case of linear sub-
Harbourne was partially supported by Simons Foundation Grant # 524858. Roé was partially supported by Spanish Mineco Grant MTM2016-75980-P and by Catalan AGAUR Grant 2017SGR585. Harbourne and Tutaj-Gasi´nska were partially supported by National Science Centre Grant 2017/26/M/ST1/00707. Szemberg was partially supported by National Science Centre Grant 2018/30/M/ST1/00148. Extended author information available on the last page of the article
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spaces of higher dimension is similar to the case of points. In particular, after fixing sufficiently many sufficiently general linear subspaces, it is expected that the conditions imposed on forms by vanishing along these subspaces will be independent. (For efficiency we slightly abuse terminology by saying that r homogeneous linear equations in a vector space of dimension s are independent if the subspace of solutions has dimension either s − r or 0.) On the other hand, it is interesting and important to understand special linear systems, i.e., non-empty systems for which the imposed conditions are dependent, or, equivalently, non-empty linear systems whose dimension is greater than expected from a naive conditions count. In the classical setup of assigned base points in the case of generic points in P2, the well-known Segre–Harbourne–Gimigl
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