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Arrangements Of Hyperplanes

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Classical Algebraic Geometry

We survey classical and recent results on symbolic powers of ideals. We focus on properties and problems of symbolic powers over regular rings, on the comparison of symbolic and regular powers, and on the combinatorics of the symbolic powers of monomial ideals. In addition, we present some new results on these aspects of the subject.

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Symbolic Powers of Ideals

Branched coverings and algebraic functions

Resurgences for ideals of special point configurations in PN coming from hyperplane arrangements

This thesis consists of six papers in algebraic geometry –all of which have close connections to combinatorics. In Paper A we consider complete smooth toric embeddings X ↪ P^N such that for a fixed positive integer k the t-th osculating space at every point has maximal dimension if and only if t ≤ k. Our main result is that this assumption is equivalent to that X ↪ P^N is associated to a Cayley polytope of order k having every edge of length at least k. This result generalizes an earlier characterisation by David Perkinson. In addition we prove that the above assumptions are equivalent to requiring that the Seshadri constant is exactly k at every point of X, generalizing a result of Atsushi Ito. In Paper B we introduce H-constants that measure the negativity of curves on blow-ups of surfaces. We relate these constants to the bounded negativity conjecture. Moreover we provide bounds on H-constants when restricting to curves which are a union of lines in the real or complex projective plane. In Paper C we study Gauss maps of order k for k > 1, which maps a point on a variety to its k-th osculating space at that point. Our main result is that as in the case k = 1, the higher order Gauss maps are finite on smooth varieties whose k-th osculating space is full-dimensional everywhere. Furthermore we provide convex geometric descriptions of these maps in the toric setting. In Paper D we classify fat point schemes on Hirzebruch surfaces whose initial sequence are of maximal or close to maximal length. The initial degree and initial sequence of such schemes are closely related to the famous Nagata conjecture. In Paper E we introduce the package LatticePolytopes for Macaulay2. The package extends the functionality of Macaulay2 for compuations in toric geometry and convex geometry. In Paper F we compute the Seshadri constant at a general point on smooth toric surfaces satisfying certain convex geometric assumptions on the associated polygons. Our computations relate the Seshadri constant at the general point with the jet seperation and unnormalised spectral values of the surfaces at hand.

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Bounded Negativity and Arrangements of Lines

The purpose of this note is to give counterexamples to the containment $I^{(3)}\subset I^2$ over the real numbers.

A counterexample to the containment $I^{(3)}\subset I^2$ over the reals

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

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

Points fattening on P^1 x P^1 and symbolic powers of bi-homogeneous ideals

The main goal of the paper is to begin a systematic study on conic-line arrangements in the complex projective plane. We show a de Bruijn-Erdos-type inequality and Hirzebruch-type inequality for a certain class of conic-line arrangements having ordinary singularities. We will also study, in detail, certain conic-line arrangements in the context of the geography of log-surfaces and free divisors in the sense of Saito.

Piotr Pokora

Papers

Bounded Negativity and Arrangements of Lines

A counterexample to the containment $I^{(3)}\subset I^2$ over the reals

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

Points fattening on P^1 x P^1 and symbolic powers of bi-homogeneous ideals

Conic-line arrangements in the complex projective plane