Bounded Negativity Conjecture Research Articles

On the Degree of Curves with Prescribed Multiplicities and Bounded Negativity

Abstract We provide a lower bound on the degree of curves of the projective plane $\mathbb {P}^2$ passing through the centers of a divisorial valuation $\nu $ of $\mathbb {P}^2$ with prescribed multiplicities, and an upper bound for the Seshadri-type constant of $\nu $, $\hat {\mu }(\nu )$, constant that is crucial in the Nagata-type valuative conjecture. We also give some results related to the bounded negativity conjecture concerning those rational surfaces having the projective plane as a relatively minimal model.

On the Bounded Negativity Conjecture and Singular Plane Curves

There are no known failures of Bounded Negativity in characteristic 0. In the light of recent work showing the Bounded Negativity Conjecture fails in positive characteristics for rational surfaces, we propose new characteristic free conjectures as a replacement. We also develop bounds on numerical characteristics of curves constraining their negativity. For example, we show that the $H$-constant of a rational curve $C$ with at most $9$ singular points satisfies $H(C)>-2$ regardless of the characteristic.

Unbounded negativity on rational surfaces in positive characteristic

Abstract We give explicit blowups of the projective plane in positive characteristic that contain smooth rational curves of arbitrarily negative self-intersection, showing that the Bounded Negativity Conjecture fails even for rational surfaces in positive characteristic. As a consequence, we show that any surface in positive characteristic admits a birational model failing the Bounded Negativity Conjecture.

Bounded negativity and Harbourne constants on ruled surfaces

Let X be a smooth projective surface and let $${\mathcal {C}}$$ be an arrangement of curves on X. The Harbourne constant of $${\mathcal {C}}$$ was defined as a way to investigate the occurrence of curves of negative self-intersection on blow ups of X. This is related to the bounded negativity conjecture which predicts that the self-intersection number of all reduced curves on a surface is bounded below by a constant. We consider a geometrically ruled surface X over a smooth curve and give lower bounds for the Harbourne constants of transversal arrangements of curves on X. We also define a global Harbourne constant as the infimum of Harbourne constants for arrangements of a specific type and give a lower bound for it.

Towards the weighted Bounded Negativity Conjecture for blow-ups of algebraic surfaces

In the present paper we focus on a weighted version of the Bounded Negativity Conjecture, which predicts that for every smooth projective surface in characteristic zero the self-intersection numbers of reduced and irreducible curves are bounded from below by a function depending on the intesection of curve with an arbitrary big and nef line bundle that is positive on the curve. We gather evidence for this conjecture by showing various bounds on the self-intersection number of curves in an algebraic surface. We focus our attention on blow-ups of algebraic surfaces, which have so far been neglected.

Weak bounded negativity conjecture

In this paper, we prove the following “weak bounded negativity conjecture”, which says that given a complex smooth projective surface X X , for any reduced curve C C in X X and integer g g , assume that the geometric genus of each component of C C is bounded from above by g g ; then the self-intersection number C 2 C^2 is bounded from below.

Some surfaces with non-polyhedral nef cones

Over algebraically closed fields of positive characteristic, C×C contains the family of curves cut out as graphs of iterates of the Frobenius morphism. Their self-intersection tends to negative infinity; hence C × C fails the bounded negativity conjecture in positive characteristic. This work arises as part of the broader question of whether C×C where C has genus ≥ 2 satisfies the bounded negativity conjecture in characteristic 0. When a surfaceX has a polyhedral nef cone, it automatically satisfies bounded negativity since all negative curves are extremal and polyhedral cones have finitely many extremal rays. We establish that this naive approach to bounded negativity fails for C×C. The original question remains open.

Curve configurations in the projective plane and their characteristic numbers

In this paper we study the concept of characteristic numbers and Chern slopes in the context of curve configurations in the real and complex projective plane. We show that some extremal line configurations inherit the same asymptotic invariants, namely asymptotic Chern slopes and asymptotic Harbourne constants which sheds some light on relations between the bounded negativity conjecture and the geography problem for surfaces of general type. We discuss some properties of Kummer extensions, especially in the context of ball-quotients. Moreover, we prove that for a certain class of smooth curve configurations in the projective plane their characterstic numbers are bounded by $8/3$.

Bounded negativity, Harbourne constants and transversal arrangements of curves

The Bounded Negativity Conjecture predicts that for every complex projective surface $X$ there exists a number $b(X)$ such that $C^2\geq -b(X)$ holds for all reduced curves $C\subset X$. For birational surfaces $f:Y\to X$ there have been introduced certain invariants (Harbourne constants) relating to the effect the numbers $b(X)$, $b(Y)$ and the complexity of the map $f$. These invariants have been studied previously when $f$ is the blowup of all singular points of an arrangement of lines in ${\mathbb P}^2$, of conics and of cubics. In the present note we extend these considerations to blowups of ${\mathbb P}^2$ at singular points of arrangements of curves of arbitrary degree $d$. We also considerably generalize and modify the approach witnessed so far and study transversal arrangements of sufficiently positive curves on arbitrary surfaces with the non-negative Kodaira dimension.

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

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 {mathbb {B}}12 configurations is a three dimensional rational variety. As a consequence we derive the existence of a three dimensional family of rational {mathbb {B}}12 configurations. On the other hand the parameter space of {mathbb {B}}15 configurations is shown to be an elliptic curve with only finitely many rational points, all corresponding to degenerate configurations. Thus, somewhat surprisingly, we conclude that there are no rational {mathbb {B}}15 configurations.

Bounded negativity of self-intersection numbers of Shimura curves in Shimura surfaces

Shimura curves on Shimura surfaces have been a candidate for counterexamples to the bounded negativity conjecture. We prove that they do not serve this purpose: there are only finitely many whose self-intersection number lies below a given bound. Previously, this result has been shown in [BHK+13] for compact Hilbert modular surfaces using the Bogomolov-Miyaoka-Yau inequality. Our approach uses equidistribution and works uniformly for all Shimura surfaces.

On Linear Harbourne Constants

In this note we compute values of global linear Harbourne constants over arbitrary fields for up to ten lines. These invariants have appeared recently in the discussions around the Bounded Negativity Conjecture. They seem to be of independent interest also from the point of view of combinatorics.

Bounded Negativity and Arrangements of Lines

The Bounded Negativity Conjecture predicts that for any smooth complex surface $X$ there exists a lower bound for the selfintersection of reduced divisors on $X$. This conjecture is open. It is also not known if the existence of such a lower bound is invariant in the birational equivalence class of $X$. In the present note we introduce certain constants $H(X)$ which measure in effect the variance of the lower bounds in the birational equivalence class of $X$. We focus on rational surfaces and relate the value of $H({\mathbb P}^2)$ to certain line arrangements. Our main result is Theorem 3.3 and the main open challenge is Problem 3.10.

On the influence of the Segre Conjecture on the Mori cone of blown-up surfaces

We propose a generalization of SHGH Conjectures to a smooth projective surface Y: the so-called Generalized Segre Conjecture. The study of linear systems on Y can be translated in terms of the Mori cone of the blow-up $$X=\mathrm{Bl }_r Y$$ at $$r$$ general points. A consequence of the Generalized Segre Conjecture is the so-called List Conjecture, a generalization of the Bounded Negativity Conjecture. Generalizing a result from [7], we prove that if the List Conjecture holds true, then a part of $$\overline{\mathrm{NE }}(X)$$ does coincide with a part of the positive cone of $$X$$ .

Negative curves on algebraic surfaces

We study curves of negative self-intersection on algebraic surfaces. We obtain results for smooth complex projective surfaces X on the number of reduced, irreducible curves C of negative self-intersection C^2. The only known examples of surfaces for which C^2 is not bounded below are in positive characteristic, and the general expectation is that no examples can arise over the complex numbers. Indeed, we show that the idea underlying the examples in positive characteristic cannot produce examples over the complex number field. The previous version of this paper claimed to give a counterexample to the Bounded Negativity Conjecture. The idea of the counterexample was to use Hecke translates of a smooth Shimura curve in order to create an infinite sequence of curves violating the Bounded Negativity Conjecture. To this end we applied Hirzebruch Proportionality to all Hecke translates, simultaneously desingularized by a version of Jaffee's Lemma which exists in the literature but which turns out to be false. Indeed, in the new version of the paper, we show that only finitely many Hecke translates of a special subvariety of a Hilbert modular surface remain smooth. This new result is based on work done jointly with Xavier Roulleau, who has been added as an author. The other results in the original posting of this paper remain unchanged.