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A containment result in $\mathbb{P}^n$ and the Chudnovsky conjectureMar 11 2016Nov 25 2016In the paper we prove the containment $I^{(nm)}\subset M^{(n-1)m}I^m$, for a radical ideal $I$ of $s$ general points in $\mathbb{P}^n$, where $s\geq 2^n$. As a corollary we get that the Chudnovsky Conjecture holds for a very general set of at least $2^n$ ... More

A containment result in $\mathbb{P}^n$ and the Chudnovsky conjectureMar 11 2016In the paper we prove the containment $I^{(nm)}\subset M^{(n-1)m}I^m$, for a radical ideal $I$ of $s$ general points in $\mathbb{P}^n$, where $s\geq 2^n$. As a corollary we get that the Chudnovsky Conjecture holds for a very general set of at least $2^n$ ... More

Asymptotic Hilbert Polynomial and limiting shapesJul 02 2014The main aim of this paper is to provide a method which allows finding limiting shapes of symbolic generic initial systems of higher-dimensional subvarieties of P^n. M. Mustata and S. Mayes established a connection between volumes of complements of limiting ... More

Asymptotic Hilbert Polynomial and a bound for Waldschmidt constantsNov 24 2015In the paper we give an upper bound for the Waldschmidt constants of the wide class of ideals. This generalizes the result obtained by Dumnicki, Harbourne, Szemberg and Tutaj-Gasinska, Adv. Math. 2014. Our bound is given by a root of a suitable derivative ... More

Symbolic powers of planar point configurationsMay 27 2012We study initial degrees of symbolic powers of ideals of arbitrary finite sets of points in the projective plane over an algebraically closed field of characteristic zero. We show, how bounds on the growth of these degrees determine the geometry of the ... More

A vanishing theorem and symbolic powers of planar point idealsFeb 04 2013The 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 ... More

Symbolic powers of planar point configurations IIApr 21 2015We study initial sequences of various configurations of planar points. We answer several questions asked in our previous paper (Symbolic powers of planar point configurations), and we extend our considerations to the asymptotic setting of Waldschmidt ... More

Lower bounds for Waldschmidt constants of generic lines in $\mathbb{P}^3$ and a Chudnovsky-type theoremMar 06 2018The Waldschmidt constant $\alphahat(I)$ of a radical ideal $I$ in the coordinate ring of $\PP^N$ measures (asymptotically) the degree of a hypersurface passing through the set defined by $I$ in $\PP^N$. Nagata's approach to the 14th Hilbert Problem was ... More

Symbolic generic initial systems of star configurationsJan 19 2014The purpose of this note is to describe limiting shapes (as introduced by Mayes) of symbolic generic initial systems of star configurations in projective spaces over a field of characteristic 0.

Restrictions on Seshadri constants on surfacesFeb 29 2016Starting with the pioneering work of Ein and Lazarsfeld restrictions on values of Seshadri constants on algebraic surfaces have been studied by many authors. In the present note we show how approximation involving continued fractions combined with recent ... More

A matrixwise approach to unexpected hypersurfacesJul 10 2019The aim of this note is to give a generalization of some results concerning unexpected hypersurfaces. Unexpected hypersurfaces occur when the actual dimension of the space of forms satisfying certain vanishing data is positive and the imposed vanishing ... More

Line arrangements with the maximal number of triple pointsJun 25 2014The 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 F over which such extremal configurations ... More

On the Sylvester-Gallai theorem for conicsNov 10 2014In the present note we give a new proof of a result due to Wiseman and Wilson which establishes an analogue of the Sylvester-Gallai theorem valid for curves of degree two. The main ingredients of the proof come from algebraic geometry. Specifically, we ... More

Harbourne constants and conic configurations on the projective planeJun 19 2015Oct 04 2015In this note we exhibit the so-called Harbourne constants which capture and measure the Bounded Negativity on various birational models of an algebraic surface. We show an estimation for Harbourne constants for conic configurations on the complex projective ... More

On integral Zariski decompositions of pseudoeffective divisors on algebraic surfacesJul 27 2015In this note we consider the problem of integrality of Zariski decompositions for pseudoeffective integral divisors on algebraic surfaces. We show that while sometimes integrality of Zariski decompositions forces all negative curves to be $(-1)$-curves, ... More

New constructions of unexpected hypersurfaces in $\mathbb{P}^n$Apr 05 2019In the paper we present new examples of unexpected varieties. The research on unexpected varieties started with a paper of Cook II, Harbourne, Migliore and Nagel and was continued in the paper of Harbourne, Migliore, Nagel and Teitler. Here we present ... More

Containment problem and combinatoricsOct 17 2017In this note we consider two configurations of twelve lines with nineteen triple points (i.e., points where three lines meet). Both of them have the same combinatorial features. In both configurations nine of twelve lines have five triple points and one ... More

On the parameter space of Böröczky configurationsJun 27 2017Feb 17 2018B\"or\"oczky configurations of lines have been recently considered in connection with the problem of the containment between symbolic and ordinary powers of ideals. Here we describe parameter families of B\"or\"oczky configurations of 13, 14, 16, 18 and ... More

Linear subspaces, symbolic powers and Nagata type conjecturesJul 05 2012Sep 30 2012Prompted by results of Guardo, Van Tuyl and the second author for lines in projective 3 space, 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 projective ... More

Counterexamples to the $I^{(3)} \subset I^2$ containmentJan 30 2013Apr 21 2015We 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.

Veneroni mapsJun 06 2019Veneroni maps are a class of birational transformations of projective spaces. This class contains the classical Cremona transformation of the plane, the cubo-cubic transformation of the space and the quatro-quartic transformation of $\mathbb{P}^4$. Their ... More

Resurgences for ideals of special point configurations in ${\bf P}^N$ coming from hyperplane arrangementsApr 19 2014Symbolic 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. Several invariants have been introduced and ... More

Points fattening on P^1 x P^1 and symbolic powers of bi-homogeneous idealsApr 21 2013We study symbolic powers of bi-homogeneous ideals of points in the Cartesian product of two projective lines and extend to this setting results on the effect of points fattening obtained by Bocci, Chiantini and Dumnicki, Szemberg, Tutaj-Gasi\'nska. We ... More

A way from the isoperimetric inequality in the plane to a Hilbert spaceSep 10 2014We will formulate and prove a generalization of the isoperimetric inequality in the plane. Using this inequality we will construct an unitary space - and in consequence - an isomorphic copy of a separable infinite dimensional Hilbert space, which will ... More

Prime numbers with a certain extremal type propertyAug 15 2014The convex hull of the subgraph of the prime counting function $x\rightarrow \pi(x)$ is a convex set, bounded from above by a graph of some piecewise affine function $x\rightarrow \epsilon(x)$. The vertices of this function form an infinite sequence of ... More

Unexpected surfaces singular on lines in $\mathbb{P}^3$Jan 11 2019We study linear systems of surfaces in $\mathbb{P}^3$ singular along general lines. Our purpose is to identify and classify special systems of such surfaces, i.e., those nonempty systems where the conditions imposed by the multiple lines are not independent. ... More

Classifying attention deficit hyperactivity disorder in children with non-linearities in actigraphyFeb 10 2019Objective This study provides an objective measure based on actigraphy for Attention Deficit Hyperactivity Disorder (ADHD) diagnosis in children. We search for motor activity features that could allow further investigation into their association with ... More