Weak bounded negativity conjecture

Hao, Feng

doi:10.1090/proc/14376

Weak bounded negativity conjecture
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Proc. Amer. Math. Soc. 147 (2019), 3233-3238 Request permission

Abstract:

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

References

Thomas Bauer, Brian Harbourne, Andreas Leopold Knutsen, Alex Küronya, Stefan Müller-Stach, Xavier Roulleau, and Tomasz Szemberg, Negative curves on algebraic surfaces, Duke Math. J. 162 (2013), no. 10, 1877–1894. MR 3079262, DOI 10.1215/00127094-2335368
Thomas Bauer, Cristiano Bocci, Susan Cooper, Sandra Di Rocco, Marcin Dumnicki, Brian Harbourne, Kelly Jabbusch, Andreas Leopold Knutsen, Alex Küronya, Rick Miranda, Joaquim Roé Hal Schenck, Tomasz Szemberg, Zach Teitler, Recent developments and open problems in linear series, Contributions to Algebraic Geometry, Impanga Lecture Notes Series, (2012), 93-140.
Thomas Bauer, A simple proof for the existence of Zariski decompositions on surfaces, J. Algebraic Geom. 18 (2009), no. 4, 789–793. MR 2524598, DOI 10.1090/S1056-3911-08-00509-2
F. A. Bogomolov, Stable vector bundles on projective surfaces, Mat. Sb. 185 (1994), no. 4, 3–26 (Russian, with Russian summary); English transl., Russian Acad. Sci. Sb. Math. 81 (1995), no. 2, 397–419. MR 1272185, DOI 10.1070/SM1995v081n02ABEH003544
Ciro Ciliberto, Brian Harbourne, Rick Miranda, Joaquim Roé, Variations on Nagata’s Conjecture, A celebration of algebraic geometry, Clay Math. Proc., 18 (2013) 185-203.
Jean-Pierre Demailly, Singular Hermitian metrics on positive line bundles, Complex algebraic varieties (Bayreuth, 1990) Lecture Notes in Math., vol. 1507, Springer, Berlin, 1992, pp. 87–104. MR 1178721, DOI 10.1007/BFb0094512
Takao Fujita, On Zariski problem, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 3, 106–110. MR 531454
Yoichi Miyaoka, The maximal number of quotient singularities on surfaces with given numerical invariants, Math. Ann. 268 (1984), no. 2, 159–171. MR 744605, DOI 10.1007/BF01456083
Oscar Zariski, The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface, Ann. of Math. (2) 76 (1962), 560–615. MR 141668, DOI 10.2307/1970376