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 ?
Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)
|
0
1
|
||||
|
|
3
|
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. |
||
|
|
You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.
|
2
|
Yes, this is possible. For an example of a Calabi-Yau threefold with such differences of curves, see my paper with Pavanelli http://arxiv.org/pdf/math/0512182.pdf. I am sure there are much simpler examples, however. |
||||||
|
|
1
|
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. ADDED LATER: They may also be chosen to be connected. The Lefschetz hyperplane theorem shows that hyperplane sections of surfaces are connected. |
|||||||||||||||
|
