arXiv subjects
Publications and source records attributed to Marcin Dumnicki.
In the paper we provide a new method of proving the existence of a hypersurface of degree $d$ in $\mathbb{P}^n$, with a general point of multiplicity $m$ and vanishing at a given set of points $Z$, by looking at weak combinatorics of a set $Z$. This method has a direct application in the theory of unexpected hypersurfaces, where many of the examples are based only on computer experiments.
The purpose of the present note is to provide a new proof ot the well-known result due to Hartshorne and Hirschowitz to the effect that general lines in projective spaces have good postulation. Our approach uses specialization to a hyperplane and thus opens door to study postulation of general codimension 2 linear subspaces in projective spaces.
We study negative curves on surfaces obtained by blowing up special configurations of points in the complex projective palne. Our main results concern the following configurations: very general points on a cubic, 3-torsion points on an elliptic curve and nine Fermat points. As a consequence of our analysis, we also show that the Bounded Negativity Conjecture holds for the surfaces we consider. The note contains also some problems for future attention.
The 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 conditions are not independent. The first instance studied were unexpected curves in the paper by Cook II, Harbourne, Migliore, Nagel. Unexpected hypersurfaces were then investigated by Bauer, Malara, Szpond and Szemberg, followed by Harbourne, Migliore, Nagel and Teitler who introduced the notion of BMSS duality and showed it holds in some cases (such as certain plane curves and, in higher dimensions, for certain cones). They ask to what extent such a duality holds in general. In this paper, working over a field of characteristic zero, we study hypersurfaces in $\mathbb{P}^n\times\mathbb{P}^n$ defined by determinants. We apply our results to unexpected hypersurfacesin the case that the actual dimension is 1 (i.e., there is a unique unexpected hypersurface). In this case, we show that a version of BMSS duality always holds, as a consequence of fundamental properties of determinants.
The 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 based on computing such constant for the set of points in $\PP^2$. Since then, these constants drew much attention, but still there are no methods to compute them (except for trivial cases). Therefore the research focuses on looking for accurate bounds for $\alphahat(I)$. In the paper we deal with $\alphahat(s)$, the Waldschmidt constant for $s$ very general lines in $\PP^3$. We prove that $\alphahat(s) \geq \lfloor\sqrt{2s-1}\rfloor$ holds for all $s$, whereas the much stronger bound $\alphahat(s) \geq \lfloor\sqrt{2.5 s}\rfloor$ holds for all $s$ but $s=4$, $7$ and $10$. We also provide an algorithm which gives even better bounds for $\alphahat(s)$, very close to the known upper bounds, which are conjecturally equal to $\alphahat(s)$ for $s$ large enough.
In this note we introduce a Waldschmidt decomposition of divisors which might be viewed as a generalization of Zariski decomposition based on the effectivity rather than the nefness of divisors. As an immediate application we prove a recursive formula providing new effective lower bounds on Waldschmidt constants of very general points in projective spaces. We use these bounds in order to verify Demailly's conjecture in a number of new cases.
In 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 $2^n$ points in $\mathbb{P}^n$.
It is well known that multi-point Seshadri constants for a small number $s$ of points in the projective plane are submaximal. It is predicted by the Nagata conjecture that their values are maximal for $s\geq 9$ points. Tackling the problem in the language of valuations one can make sense of $s$ points for any positive real $s\geq 1$. We show somewhat surprisingly that a Nagata-type conjecture should be valid for $s\geq 8+1/36$ points and we compute explicitly all Seshadri constants (expressed here as the asymptotic maximal vanishing element) for $s\leq 7+1/9$.
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 $s$ of the form $p^{2r}+p^r+1$ for any prime number $p$ and also for all values of $s$ up to $31$. This extends considerably earlier results of the third author.
In 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 of a certain polynomial associated with the asymptotic Hilbert polynomial.
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.
We 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 constants. We introduce the concept of Bezout Decomposition which might be of independent interest.
In 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 use Cremona transformation of the projective plane and Hirzebruch inequality.
The 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 shapes and the asymptotic multiplicity for ideals of points. In the paper we prove a generalization of this fact to higher-dimensional sets.
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 F over which such extremal configurations exist. We show that there does not exist a field admitting a configuration of 11 lines with 17 triple points, even though such a configuration is allowed combinatorially. Finally, we present an infinite series of configurations which have a high number of triple intersection points.
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. Several invariants have been introduced and studied in the latter context, including the resurgence and asymptotic resurgence. There have been exciting new developments in this area recently. It had been expected for several years that $I^{Nr-N+1}\subseteq I^r$ should hold for the ideal $I$ of any finite set of points in ${\bf P}^N$ for all $r>0$, but in the last year various counterexamples have now been constructed, 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.
The 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.
We 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-Gasińska. We prove a Chudnovsky-type theorem for bi-homogeneous ideals and apply it to classification of configurations of points with minimal or no fattening effect. We hope that the ideas developed in this project will find further algebraic and geometric applications e.g. to study similar problems on arbitrary surfaces.