Constructing non-Mori Dream Spaces from negative curves

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

“…From the results of GonzálezAnaya-González-Karu [7], it can be inferred that R ′ s (p a,b,c ) tends to be finitely generated when |d 1 P a,b,c | is close to |P ϕ |, and R ′ s (p a,b,c ) tends to be infinitely generated when |d 1 P a,b,c | is close to r 2 /2. Therefore it is natural to ask: Is R ′ s (p ϕ ) a Noetherian ring?…”

“…Suppose that the characteristic of k is 0. In GonzálezAnaya-González-Karu [7], they constructed two distinct r-ncts ϕ r and ϕ ′ r over k for each r ≥ 3. Here P ϕr is a triangle with vertices (−1, −1), (r − 1, 0) and (0, r − 1).…”

“…In this case, the extended symbolic Rees ring R s (p a,b,c )[t −1 ] is the Cox ring of the blow-up Y ∆ a,b,c of the weighted projective surface X ∆ a,b,c = Proj(k[x, y, z]) at the point (1,1) in the torus. Many commutative algebraists and algebraic geometers studied them and gave many results ( [13], [3], [9], [10], [6], [7], [17], [18] etc.). Finite generation of R s (p a,b,c ) depends on a, b, c and the characteristic of k. There are many examples of finitely generated R s (p a,b,c ).…”

Effective cone of the blowup of the symmetric product of a curve

“…Deciding when the (pseudo)effective cone of S is polyhedral is an open problem even when S is a toric surface. For instance, if the effective cone of the blowup of the weighted projective plane P(a, b, c) at a general point is not closed, then Nagata's conjecture holds for abc points in P 2 ; see [4] and [8][9][10][11] for recent results on blowups of weighted projective planes. In [2] it has been shown that there exist toric surfaces whose blowup at a general point has nonpolyhedral pseudoeffective cone.…”

Mori dreamness of blowups of weighted projective planes

“…In [2,4,6,9], the authors give examples of X(a, b, c) that are Mori dream spaces. In [3,5], there are examples of X(a, b, c) that are not Mori dream spaces.…”