The maximum genus problem asks for the largest arithmetic genus g(d,s) achievable by a locally Cohen-Macaulay curve of degree d in complex projective 3-space that lies on no surface of degree less than s. This work proves the conjectured bound is sharp when d = s or d >= 2s-1. The key step is a proof of the Beorchia-Lella-Schlesinger conjecture on the initial ideals of certain weighted homogeneous forms in a non-standard graded polynomial ring: understanding these monomial initial ideals fixes the relevant Hilbert function and therefore the extremal genus, tying commutative-algebra (Groebner degeneration, weighted gradings) to the classical geometry of space curves and liaison theory.
For a locally Cohen-Macaulay curve in complex projective 3-space of degree d that lies on no surface of degree less than s, the maximum genus problem predicts the largest possible arithmetic genus g(…