Bounded negativity, Harbourne constants and transversal arrangements of curves

Pokora, Piotr; Roulleau, Xavier; Szemberg, Tomasz

doi:10.5802/aif.3149

Bounded negativity, Harbourne constants and transversal arrangements of curves
[Négativité bornée, constantes de Harbourne et arrangements transverses de courbes]

Pokora, Piotr ¹ ; Roulleau, Xavier ² ; Szemberg, Tomasz ¹

¹ Department of Mathematics Pedagogical University of Cracow Podchorążych 2 30-084 Kraków (Poland)
² Laboratoire de Mathématiques et Applications Université de Poitiers, UMR CNRS 7348 Téléport 2 - BP 30179 86962 Futuroscope Chasseneuil (France)

Annales de l'Institut Fourier, Tome 67 (2017) no. 6, pp. 2719-2735.

Résumé
Abstract

La conjecture de la négativité bornée prédit que pour toute surface complexe projective $X$ , il existe un nombre $b (X)$ tel que l’inégalité $C^{2} \geq - b (X)$ ait lieu pour toute courbe réduite $C \subset X$ . Pour un morphisme birationnel $f : Y \to X$ , certains invariants (les constantes de Harbourne) ont été introduits afin de relier les nombres $b (X)$ et $b (Y)$ à la complexité de $f$ . Ces invariants ont été étudiés quand $f$ est l’éclatement en tous les points singuliers d’un arrangement de droites, de coniques et de cubiques. Dans cette note, nous étendons ces considérations aux éclatements de $ℙ^{2}$ aux points singuliers d’arrangements de courbes de degré arbitraire $d$ . Le résultat principal dans cette direction est le théorème B. Ensuite, nous généralisons considérablement et modifions l’approche usuelle afin d’étudier les arrangements transverses de courbes suffisamment positives sur n’importe quelle surface ayant dimension de Kodaira positive ou nulle. Le principal résulat obtenu dans ce cadre général est le théorème A.

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 when $f$ is the blowup of all singular points of an arrangement of lines in $ℙ^{2}$ , of conics and of cubics. In the present note we extend these considerations to blowups of $ℙ^{2}$ at singular points of arrangements of curves of arbitrary degree $d$ . The main result in this direction is stated in Theorem B. 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. The main result obtained in this general setting is presented in Theorem A.

Reçu le : 2016-02-12
Révisé le : 2016-06-30
Accepté le : 2017-03-15
Publié le : 2017-12-14

DOI : 10.5802/aif.3149

Classification : 14C20, 14J70
Keywords: curve arrangements, algebraic surfaces, Miyaoka inequality, blow-ups, negative curves, bounded negativity conjecture
Mot clés : arrangements de courbes, surfaces algébriques, inégalité de Miyaoka, courbes négatives, conjecture de la négativité Bornée

Affiliations des auteurs :

Pokora, Piotr ¹ ; Roulleau, Xavier ² ; Szemberg, Tomasz ¹

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Bounded negativity, {Harbourne} constants and transversal arrangements of curves},
     journal = {Annales de l'Institut Fourier},
     pages = {2719--2735},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Pokora, Piotr; Roulleau, Xavier; Szemberg, Tomasz. Bounded negativity, Harbourne constants and transversal arrangements of curves. Annales de l'Institut Fourier, Tome 67 (2017) no. 6, pp. 2719-2735. doi : 10.5802/aif.3149. https://aif.centre-mersenne.org/articles/10.5802/aif.3149/

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