By Dirk Kussin

In those notes the writer investigates noncommutative gentle projective curves of genus 0, also known as unheard of curves. As a major consequence he indicates that every such curve X admits, as much as a few weighting, a projective coordinate algebra that's a no longer unavoidably commutative graded factorial area R within the feel of Chatters and Jordan. additionally, there's a typical bijection among the issues of X and the homogeneous best beliefs of top one in R, and those leading beliefs are significant in a robust feel

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Equivalently, the norm form of pure quaternions −aX 2 − bY 2 + abZ 2 is anisotropic over k. Let M be the bimodule k FF . We have shown in [54] that the small preprojective algebra is given by Π(L, τ − ) k[X, Y, Z]/(−aX 2 − bY 2 + abZ 2 ). We will see later that τ − is the only eﬃcient automorphism in this case. Note that the factoriality of this algebra was already known from a theorem of P. Samuel [99], we refer to [33, Prop. 5]. It is interesting that the bimodule k FF given by noncommutative data gives a commutative orbit algebra.

Let K = Z ∩ Fix(α). 7. The centre of R = F [X; Y, α] is given by K[X, uY r ]. The homogeneous prime elements in R are (up to multiplication with a unit) X, Y and the homogeneous prime elements in K[X, uY r ], which are polynomials in X r and uY r with coeﬃcients lying in K. Proof. 5. ) It follows in particular that (up to multiplication with a unit) every homogeneous prime element in R except Y is central. ) Note that for example the central elements of the form aX r + buY r with a, b ∈ K ∗ are prime in R.

By [89] there are α, δ : F −→ F such that for all f ∈ F the formula πy f = δ(f )πx + α(f )πy holds, where α is a k-automorphism of F and δ is an (α, 1)-derivation on F . Since dimR0 Rn = n + 1, it is easy to see that the n + 1 elements πxn , πxn−1 πy , πxn−2 πy2 , . . , πx πyn−1 , πyn form a R0 -basis of Rn for each natural number n. Denote by F [X; Y, α, δ] the skew polynomial ring in two variables, where every element is expressible uniquely in the form i, j fij X i Y j with fij ∈ F (that is, as left polynomial) and such that X is central and for all f ∈ F we have Y f = δ(f )X + α(f )Y.