Algebraic surfaces of general type - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T22:30:57Z http://mathoverflow.net/feeds/question/102670 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/102670/algebraic-surfaces-of-general-type Algebraic surfaces of general type roger-yang 2012-07-19T14:16:47Z 2012-07-19T14:50:48Z <p>Question. Are there smooth complex surfaces of general type with an irregularity $q = 1$ and Euler characteristic $3$. </p> <p>If the answer is yes, what is known about the geometry of such surfaces? Are there some explicit constructions? </p> http://mathoverflow.net/questions/102670/algebraic-surfaces-of-general-type/102672#102672 Answer by Francesco Polizzi for Algebraic surfaces of general type Francesco Polizzi 2012-07-19T14:30:52Z 2012-07-19T14:50:48Z <p>If you mean <em>holomorphic</em> Euler characteristic equal to $3$, i.e. $p_g=3$, then the answer is <em>yes</em>.</p> <p>For some explicit constructions, look at the paper by Takahashi </p> <p><a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.tmj/1178224978" rel="nofollow">Certain algebraic surfaces of general type with irregularity one and their canonical mappings</a>, Tohoku Math. J. (2) Volume <strong>50</strong>, Number 2 (1998), 261-290. </p> <p>For any value of $p_g \geq 2$, the author shows the existence of minimal surfaces of general type with $K^2=3p_g$ and $q=1$ and studies their canonical mappings.</p> <p>These surfaces are the minimal resolution of a relative quartic hypersurface (having at most rational double points as singularities) in a $\mathbf{P}^2$-bundle over an elliptic curve.</p> <p>If you mean instead <em>topological</em> Euler characteristic, Noether formula gives $$K^2=12 \chi(\mathcal{O}) -3.$$ </p> <p>On the other hand, by Bogomolov-Miyaoka-Yau inequality one has $K^2 \leq 9 \chi(\mathcal{O})$, so the only possibility is $\chi(\mathcal{O})=1$, i.e. $p_g=q=1$, $K^2=9$.</p> <p>As far as I know, the existence of such a surface was announced by Cartwright and Steger. They used a ball-quotient construction (based also on computer calculations), similar to the one used by Prasad and Yeung in order to classify the fake projective planes. I do not think that a more explicit geometric construction is currently known. </p>