Justyna Szpond

On the containment problem

The purpose of this note is to provide an overview of the containment problem for symbolic and ordinary powers of homogeneous ideals, related conjectures and examples. We focus here on ideals with zero dimensional support. This is an area of ongoing active research. We conclude the note with a list of potential promising paths of further research.

A counterexample to the containment I(3) ⊂ I2 over the reals

The purpose of this note is to give counterexamples to the con- tainment I (3) I 2 over the real numbers.

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

From Pappus Theorem to parameter spaces of some extremal line point configurations and applications

1

Symbolic generic initial systems of star configurations

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On absolute linear Harbourne constants

In the present work we study parameter spaces of two line point configurations introduced by Böröczky. These configurations are extremal from the point of view of the Dirac–Motzkin Conjecture settled recently by Green and Tao (Discrete Comput Geom 50:409–468, 2013). They have appeared also recently in commutative algebra in connection with the containment problem for symbolic and ordinary powers of homogeneous ideals (Dumnicki et al. in J Algebra 393:24–29, 2013) and in algebraic geometry in considerations revolving around the Bounded Negativity Conjecture (Bauer et al. in Duke Math J 162:1877–1894, 2013). We show that the parameter space of what we call

On the postulation of lines and a fat line

{\mathbb {B}}12

Line arrangements with the maximal number of triple points

B12 configurations is a three dimensional rational variety. As a consequence we derive the existence of a three dimensional family of rational