difference of curve classes - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T08:19:10Z http://mathoverflow.net/feeds/question/110615 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110615/difference-of-curve-classes difference of curve classes Mohammad F.Tehrani 2012-10-25T04:15:58Z 2012-10-26T20:11:43Z <p>Let \$X\$ be a smooth protective variety, or just a smooth Kahler manifold. Is it possible to have two curves \$C_1\$ and \$C_2\$ in \$X\$ such that their difference in \$H_2(X,\mathbb{Z})\$ is a non-trivial torsion class ?</p> http://mathoverflow.net/questions/110615/difference-of-curve-classes/110620#110620 Answer by Mark Gross for difference of curve classes Mark Gross 2012-10-25T04:59:53Z 2012-10-25T04:59:53Z <p>Yes, this is possible. For an example of a Calabi-Yau threefold with such differences of curves, see my paper with Pavanelli <a href="http://arxiv.org/pdf/math/0512182.pdf" rel="nofollow">http://arxiv.org/pdf/math/0512182.pdf</a>. I am sure there are much simpler examples, however.</p> http://mathoverflow.net/questions/110615/difference-of-curve-classes/110626#110626 Answer by Jim Bryan for difference of curve classes Jim Bryan 2012-10-25T07:47:48Z 2012-10-25T07:47:48Z <p>This is off the top of my head, but I think that the canonical class of the Enriques surface is a torsion class given by the difference of curves. Every Enriques surface is obtained from a rational elliptic surface by performing log-transforms on two of the elliptic fibers. The class \$F_1 + F_2 - F\$, where \$F_i\$ are the transformed fibers and \$F\$ is a generic fiber, is then 2-torsion. </p> http://mathoverflow.net/questions/110615/difference-of-curve-classes/110629#110629 Answer by Philip Engel for difference of curve classes Philip Engel 2012-10-25T08:19:36Z 2012-10-26T20:11:43Z <p>Every divisor class \$D\$ on a surface is the difference of two smooth, connected curves. Choose a very ample divisor \$A\$ and an \$n\$ so that \$D+nA\$ is also very ample. Then \$(D+nA)-nA=D\$ so \$D\$ is the difference of two curves. They may be chosen smoothly by Bertini's Theorem.</p> <p>ADDED LATER: They may also be chosen to be connected. The Lefschetz hyperplane theorem shows that hyperplane sections of surfaces are connected.</p>