Generic dimension and non-smoothability conjecture for homogeneous curves

Let $L_r^n$ denote the homogeneous curve considered in the paper, and let $T^1_l$ be the weight-$l$ component of its first-order deformation space. Let the dimensions of these components be bounded by Propositions 1 and 2 of the paper, and let the range in which non-smoothability is asserted be that of the paper's non-smoothability proposition.

Generic dimension and non-smoothability conjecture. For generic $L_r^n$, equality holds in the stated bounds for $\dim T^1_l$ when $l>0$ and for $\dim T^1_{-1}$, with some exceptions for low values of $r$ where the existence of smoothings forces $T^1_{-1}$ to be nonzero. Moreover, generic $L_r^n$ is not smoothable in the range of the non-smoothability proposition.

The claim predicts the generic dimensions of the relevant graded deformation spaces and uses the vanishing of negative-weight deformations to identify nonsmoothable homogeneous curves. The precise low-$r$ exceptions and the asserted nonsmoothable range are determined by the propositions referenced in the source.