Equations of negative curves of blow-ups of Ehrhart rings of rational convex polygons

Finite generation of the symbolic Rees ring a space monomial prime ideal 3-dimensional weighted polynomial is very interesting problem. Negative curves play important roles in finite these rings. We are interested structure negative curve. shall prove that rational many cases. also see Cox blow-up toric variety at point (1,1,…,1) coincides with extended an ring. For example, Roberts' second counterexample to Cowsik's question (and Hilbert's 14th problem) some normal projective (Remark 2.7).