Numerically rigid nef divisor - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T20:44:13Z http://mathoverflow.net/feeds/question/46224 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/46224/numerically-rigid-nef-divisor Numerically rigid nef divisor Oren 2010-11-16T11:27:04Z 2010-11-16T14:02:42Z <p>Is it possible to find an example of an \$\mathbb{R}\$-Cartier divisor \$D\$ on an irreducible variety \$X\$ that is non-trivial, nef, effective and numerically rigid? </p> <p>By "numerically rigid" I mean that if \$E\$ is another \$\mathbb{R}\$-Cartier effective divisor such that \$E\$ is numerically equivalent to \$D\$ then \$D=E\$.</p> <p>For curves this clearly cannot be the case, since an effective non-trivial divisor is always ample.</p> http://mathoverflow.net/questions/46224/numerically-rigid-nef-divisor/46225#46225 Answer by Francesco Polizzi for Numerically rigid nef divisor Francesco Polizzi 2010-11-16T11:53:32Z 2010-11-16T14:02:42Z <p>Take a minimal surface \$S\$ of general type with \$p_g=1\$, \$q=0\$ and zero torsion.</p> <p>Then \$S\$ contains a unique effective canonical curve \$K\$, which is nef and numerically rigid.</p> <p>In fact, since \$q=0\$ and there is no torsion, we have \$\textrm{Pic}^0(S)=0\$, the Neron - Severi group \$\textrm{NS}(S)\$ coincides with the Picard group \$\textrm{Pic}(S)\$ and any two numerically equivalent divisors on \$S\$ are linearly equivalent. </p> <p>Examples of these surfaces, with \$K^2=2\$, are given in the paper of Debarre and Catanese</p> <p>"Surfaces with \$K^2=2\$, \$p_g=1\$, \$q=0\$",</p> <p>J. reine angew. Math. 395 (1989), 1-55. </p>