Marcin Dumnicki

Counterexamples to the I(3)⊂I2 containment

We show that in general the third symbolic power of a radical ideal of points in the complex projective plane is not contained in the second usual power of that ideal. This answers in negative a question asked by Huneke and generalized by Harbourne.

Counterexamples to the

We show that in general the third symbolic power of a radical ideal of points in the complex projective plane is not contained in the second usual power of that ideal. This answers in negative a question asked by Huneke and generalized by Harbourne.

I^{(3)} \subset I^2

Abstract Inspired by results of Guardo, Van Tuyl and the second author for lines in P 3 , we develop asymptotic upper bounds for the least degree of a homogeneous form vanishing to order at least m on a union of disjoint r-dimensional planes in P n for n ⩾ 2 r + 1 . These considerations lead to new conjectures that suggest that the well known conjecture of Nagata for points in P 2 is not an exotic statement but rather a manifestation of a much more general phenomenon which seems to have been overlooked so far.

containment

The main goal of this paper is to present an algorithm bounding the dimension of a linear system of plane curves of given degree (or monomial basis) with multiple points in general position. As a result we prove the Harbourne-Hirschowitz conjecture when the multiplicities of base points are bounded by 11. This gives a partial answer to the question of when bivariate polynomial interpolation is possible.

Linear subspaces, symbolic powers and Nagata type conjectures

Abstract Symbolic powers of ideals have attracted interest in commutative algebra and algebraic geometry for many years, with a notable recent focus on containment relations between symbolic powers and ordinary powers; see for example [3] , [7] , [13] , [16] , [18] , [19] , [20] to cite just a few. Several invariants have been introduced and studied in the latter context, including the resurgence and asymptotic resurgence [3] , [15] . There have been exciting new developments in this area recently. It had been expected for several years that I ( N r − N + 1 ) ⊆ I r should hold for the ideal I of any finite set of points in P N for all r > 0 , but in the last year various counterexamples have now been constructed (see [11] , [17] , [8] ), all involving point sets coming from hyperplane arrangements. In the present work, we compute their resurgences and obtain in particular the first examples where the resurgence and the asymptotic resurgence are not equal.

New effective bounds on the dimension of a linear system in P2

In the week 3--9, October 2010, the Mathematisches Forschungsinstitut at Oberwolfach hosted a mini workshop Linear Series on Algebraic Varieties. These notes contain a variety of interesting problems which motivated the participants prior to the event, and examples, results and further problems which grew out of discussions during and shortly after the workshop. A lot of arguments presented here are scattered in the literature or constitute folklore. It was one of our aims to have a usable and easily accessible collection of examples and results.

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

The purpose of this note is twofold. We present first a vanishing theorem for families of linear series with base ideal being a fat points ideal. We apply then this result in order to give a partial proof of a conjecture raised by Bocci, Harbourne and Huneke concerning containment relations between ordinary and symbolic powers of planar point ideals.

Recent developments and open problems in linear series

Abstract The purpose of this note is to describe limiting shapes of symbolic generic initial systems of star configurations in projective spaces P n over a field K of characteristic 0.

A vanishing theorem and symbolic powers of planar point ideals

In the present note we study absolute linear Harbourne constants. These are invariants which were introduced in order to relate the lower bounds on the selfintersection of negative curves on birationally equivalent surfaces to the complexity of the birational map between them. We provide various lower and upper bounds on Harbourne constants and give their values for the number of lines

Symbolic generic initial systems of star configurations

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