Characterization of equality of Betti numbers for affine and projective monomial curves

Determine necessary and sufficient conditions under which the Betti sequences of the coordinate rings k[C_1] and k[C] are identical, where C ⊂ ℙ_k^n is the projective monomial curve parametrically defined by x_i = u^{a_i} v^{d−a_i} for i ∈ {0,…,n} with 0 = a_0 < a_1 < ⋯ < a_n = d, and C_1 ⊂ 𝔸_k^n is its affine chart defined by x_i = t^{a_i} for i ∈ {1,…,n}.

Background

The paper introduces a combinatorial, Gröbner-free sufficient criterion ensuring equality of the Betti numbers between the coordinate rings of a projective monomial curve C and its affine chart C_1, via an isomorphism of poset structures on corresponding Apéry sets. This condition is sufficient but, as shown by examples, not necessary.

The authors’ main theorem (Theorem 3.1 in the paper’s numbering) gives a positive criterion, but a complete characterization is still missing. Hence they explicitly pose the problem of fully characterizing when the Betti sequences of k[C] and k[C_1] coincide.