Unexpected surfaces singular on lines in $${\mathbb {P}}^{3}$$

We study linear systems of surfaces in $${\mathbb {P}}^3$$ P 3 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 $${\mathbb {P}}^3$$ P 3 . As an application we find Waldschmidt constants of six general lines in $${\mathbb {P}}^3$$ P 3 and an upper bound for this invariant for seven general lines.

On the configurations of codimension two linear subspaces of ℙN with the Waldschmidt constant less than 2

The purpose of this note is to generalize a result of [M. Dumnicki, T. Szemberg and H. Tutaj-Gasińska, Symbolic powers of planar point configurations II, J. Pure Appl. Alg. 220 (2016) 2001–2016] to higher-dimensional projective spaces and classify all configurations of [Formula: see text]-planes [Formula: see text] in [Formula: see text] with the Waldschmidt constants less than two. We also determine some numerical and algebraic invariants of the defining ideals [Formula: see text] of these classes of configurations, i.e. the resurgence, the minimal free resolution and the regularity of [Formula: see text], as well as the Hilbert function of [Formula: see text].

Negative Curves on Symmetric Blowups of the Projective Plane, Resurgences, and Waldschmidt Constants

The Klein and Wiman configurations are highly symmetric configurations of lines in the projective plane arising from complex reflection groups. One noteworthy property of these configurations is that all the singularities of the configuration have multiplicity at least 3. In this paper we study the surface X obtained by blowing up $\mathbb{P}^{2}$ in the singular points of one of these line configurations. We study invariant curves on X in detail, with a particular emphasis on curves of negative self-intersection. We use the representation theory of the stabilizers of the singular points to discover several invariant curves of negative self-intersection on X, and use these curves to study Nagata-type questions for linear series on X. The homogeneous ideal I of the collection of points in the configuration is an example of an ideal where the symbolic cube of the ideal is not contained in the square of the ideal; ideals with this property are seemingly quite rare. The resurgence and asymptotic resurgence are invariants which were introduced to measure such failures of containment. We use our knowledge of negative curves on X to compute the resurgence of I exactly. We also compute the asymptotic resurgence and Waldschmidt constant exactly in the case of the Wiman configuration of lines, and provide estimates on both for the Klein configuration.

Initial sequences and Waldschmidt constants of planar point configurations

The purpose of this work is to extend the classification of planar point configurations with low Waldschmidt constants initiated in [M. Dumnicki, T. Szemberg and H. Tutaj-Gasińska, Symbolic powers of planar point configurations II, J. Pure Appl. Algebra 220 (2016) 2001–2016] and continued in [M. Mosakhani and H. Haghighi, On the configurations of points in [Formula: see text] with the Waldschmidt constant equal to two, J. Pure Appl. Algebra 220 (2016) 3821–3825] for all values less than [Formula: see text]. As a consequence, we prove a conjecture of Dumnicki, Szemberg and Tutaj-Gasińska concerning initial sequences with low first differences.

Waldschmidt constants for Stanley–Reisner ideals of a class of simplicial complexes

We study the symbolic powers of the Stanley–Reisner ideal [Formula: see text] of a bipyramid [Formula: see text] over a [Formula: see text]-gon [Formula: see text]. Using a combinatorial approach, based on analysis of subtrees in [Formula: see text] we compute the Waldschmidt constant of [Formula: see text].