Łucja Farnik

Restrictions on Seshadri constants on surfaces

Starting with the pioneering work of Ein and Lazarsfeld [9] restrictions on values of Seshadri constants on algebraic surfaces have been studied by many authors [2,5,10,12,18,20,22,24]. In the present note we show how approximation involving continued fractions combined with recent results of Kuronya and Lozovanu on Okounkov bodies of line bundles on surfaces [13,14] lead to effective statements considerably restricting possible values of Seshadri constants. These results in turn provide strong additional evidence to a conjecture governing the Seshadri constants on algebraic surfaces with Picard number

A note on Seshadri constants of line bundles on hyperelliptic surfaces

1

Initial sequences and Waldschmidt constants of planar point configurations

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A note on k-very ampleness of line bundles on general blow-ups of hyperelliptic surfaces

We study the Seshadri constants of ample line bundles on hyperelliptic surfaces. We obtain new lower bounds and compute the exact values of the Seshadri constants in some cases. Our approach uses results of Serrano (Math. Z. 203:527–533, 1990), Harbourne and Roé (J. Pure Appl. Alg. 212:616–627, 2008), Bastianelli (Manuscripta Math. 130:113–120, 2009), Knutsen, Syzdek and Szemberg (Math. Res. Lett. 16:711–719, 2009).

Line arrangements with the maximal number of triple points

The purpose of this work is to extend the classification of planar point configurations with low Waldschmidt constants for all values less than

Containment problem and combinatorics

5/2

On k-Jet Ampleness of Line Bundles on Hyperelliptic Surfaces

. As a consequence we prove a conjecture of Dumnicki, Szemberg and Tutaj-Gasinska concerning initial sequences with low first differences.

Asymptotic Hilbert polynomial and a bound for Waldschmidt constants

We study k-very ampleness of line bundles on blow-ups of hyperelliptic surfaces at r very general points. We obtain a numerical condition on the number of points for which a line bundle on the blow-up of a hyperelliptic surface at these r points gives an embedding of order k.

On the Sylvester-Gallai theorem for conics

The purpose of this note is to study configurations of lines in projective planes over arbitrary fields having the maximal number of intersection points where three lines meet. We give precise conditions on ground fields