Stability conjecture for minimal relations of canonical rings on the projective line

Let $D$ be a

$$-divisor on$$

, supported at points $P_1,,P_n$, and let $S_D$ be its canonical ring. A minimal relation means a relation in a minimal presentation of $S_D$ by generators. The points and coefficients may vary as in the paper's setup. Minimal-relation stability conjecture. (a) If $n\leq 5$, the degrees of the minimal relations are stable. (b) For all $n$, the degrees of the minimal relations are independent of the points $P_i$ when $(P_1,\ldots,P_n)$ lies outside the locus $X_n$ of exceptional generation in Theorem (b). The conjecture is motivated by numerical data; unlike a Gröbner basis, minimal relations are intrinsic and do not depend on a monomial ordering or generating set. Stability of the minimal relations remains open, even though stability of suitable Gröbner bases is proved for $n\leq 4$.