Surfaces containing curves of arbitrarily negative self-intersection - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T21:12:52Z http://mathoverflow.net/feeds/question/63986 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/63986/surfaces-containing-curves-of-arbitrarily-negative-self-intersection Surfaces containing curves of arbitrarily negative self-intersection Olivier Benoist 2011-05-05T11:57:02Z 2012-04-05T12:39:15Z <p>Olivier Wittenberg and I are curious about the following :</p> <p>Let $S$ be a smooth projective complex surface. Are the self-intersection numbers of integral curves on $S$ always bounded below ? Or can $S$ contain integral curves with arbitrarily high negative self-intersection ?</p> http://mathoverflow.net/questions/63986/surfaces-containing-curves-of-arbitrarily-negative-self-intersection/63990#63990 Answer by Artie Prendergast-Smith for Surfaces containing curves of arbitrarily negative self-intersection Artie Prendergast-Smith 2011-05-05T12:14:43Z 2012-04-05T12:39:15Z <p>In characteristic 0 this seems to be open, but conjecturally true: a reference is Conjecture I.2.1 in <a href="http://arxiv.org/abs/0907.4151" rel="nofollow">this paper by Harbourne</a>. As mentioned in that paper, it is definitely false in positive characteristic, as one might expect.</p> <p><strong>Update:</strong> Surprisingly, this long-standing conjecture has recently been disproved! Theorem A in <a href="http://arxiv.org/abs/1109.1881" rel="nofollow">this paper</a> (by many authors) says the following:</p> <p><em>Theorem A: There exists a smooth projective complex surface containing a sequence of negative curves whose self-intersections tend to $-\infty$.</em></p> <p>The counterexamples are related to Hilbert modular surfaces. </p> <p>Let me also note that in the same paper the authors prove the following complementary theorem:</p> <p><em>Theorem B: For every integer $m>0$, there exists smooth projective complex surfaces containing infinitely many curves of self-intersection $-m$.</em></p> <p><strong>Update 2 (04/12):</strong> As John L. points out, now the authors have retracted the claimed Theorem A above. So the Bounded Negativity Conjecture is back on the cards.</p> http://mathoverflow.net/questions/63986/surfaces-containing-curves-of-arbitrarily-negative-self-intersection/63992#63992 Answer by J.C. Ottem for Surfaces containing curves of arbitrarily negative self-intersection J.C. Ottem 2011-05-05T12:17:40Z 2011-05-05T12:48:08Z <p>It is a folklore conjecture that surfaces in characteristic zero has bounded negativity. For a nice account of this problem and references, see the two survey articles</p> <p><a href="http://arxiv.org/abs/0907.4151%20" rel="nofollow">Global aspects of the geometry of surfaces</a> by Harbourne, and</p> <p><a href="http://arxiv.org/abs/1101.4363" rel="nofollow">Recent developments and open problems in linear series</a> by Bauer et al.</p> <p>In positive characteristic however, the situation is different and there is a nice counterexample due to Kollar (taken from the 2nd paper above):</p> <p>Let $C$ be a smooth curve of genus $g\geq 2$ defined over a field of characteristic $p>0$ and let $X$ be the product surface $X=C\times C$. The graph $\Gamma_q$ of the Frobenius morphism defined by taking $q=p^r$--th powers is a smooth curve of genus $g$ and self-intersection $\Gamma_q^2=q(2-2g)$. With $r$ going to infinity, we obtain a sequence of smooth curves of fixed genus with self-intersection going to minus infinity.</p>