Algebraic curves and Hilbert's fourteenth problem (Q1358389)

Negative answer has been obtained to the question raised by Hilbert (the fourteenth problem): Is the algebra of invariants finitely generated with respect to the linear action of a linear algebraic group on a vector space? But it still remains unclear for what groups the algebra of invariants is finitely generated. The positive results associated with the Hilbert fourteenth problem are applied to the theory of algebraic curves. The generalized construction recently proposed by Steinberg is useful in studying from a common standpoint the examples of linear operations whose algebras of invariants are not finitely generated. Applications of the construction in converse problems reveals that some algebra of forms is finitely generated in \(\mathbb{P}^n\).