On the Cox ring of a weighted projective plane blown-up at a point (Javier Gonzalez Anaya, HMC)

The Cox ring of a projective variety is the ring of all its meromorphic functions, together with a grading of geometric origin. Determining whether this ring is finitely generated is a challenging task, even for simple examples. In this talk, we will discuss our efforts to tackle this problem for a specific class of varieties, known as blow-ups of weighted projective planes (WPP). Through the lens of toric geometry, a WPP is characterized by a rational plane triangle. This allows us to reinterpret the problem combinatorially and show that the solution often emerges from a parameter space of such triangles. This is joint work with José Luis González and Kalle Karu.