The harmonic algebra of a lattice polytope is identified with the associated graded semigroup algebra and with a blowup section ring quotient, yielding lattice triangles whose harmonic algebra is not finitely generated.
Some non-finitely generated Cox rings.Compositio Mathematica, 152(5):984–996, 2016
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Graded Ehrhart theory and toric geometry
The harmonic algebra of a lattice polytope is identified with the associated graded semigroup algebra and with a blowup section ring quotient, yielding lattice triangles whose harmonic algebra is not finitely generated.