FoCM 2014 conference

Workshop C1 - Computational Algebraic Geometry

No date set

## Cox rings of some big rational surfaces

### UNIVERSIDAD DE LOS ANDES, COLOMBIA   -   jd.bastidas251@uniandes.edu.co

Smooth rational surfaces with big anticanonical class are known to have finitely generated Cox ring, but an explicit set of generators and their relations is not known. For a family of these surfaces, obtained as blow-ups of $\mathbb{P}^2$ at special point configurations, we prove that the Cox ring is generated by sections supported on negative curves. For some of these configurations we have a conjectural description of the relations. We believe they are generated by quadrics, and for a given degree we can decide computationally whether there are or there are not new relations in this degree. Some of our results, for both generators and relations, are computer-assisted proofs using Macaulay2. These results are joint work with Mauricio Velasco.

Joint work with MAURICIO VELASCO (UNIVERSIDAD DE LOS ANDES, COLOMBIA).