The dimension conjecture for loci of good determinantal schemes

Let $W(\underline{b};\underline{a})$ denote the locus of good determinantal schemes in $\mathbb{P}^{n}$ of codimension $c$ defined by the maximal minors of a homogeneous matrix with degree data $\underline{b}$ and $\underline{a}$. Let

$$\lambda_c:= \sum_{i,j} \binom{a_i-b_j+n}{n} + \sum_{i,j} \binom{b_j-a_i+n}{n} - \sum _{i,j} \binom{a_i-a_j+n}{n}- \sum _{i,j} \binom{b_i-b_j+n}{n} + 1.$$

Given integers $a_0\le a_1\le \cdots \le a_{t+c-2}$ and $b_1\le\cdots\le b_t$, set $\ell_i:=\sum_{j=0}^{t+i-2}a_j-\sum_{k=1}^tb_k$ and $h_{i-3}:=2a_{t+i-2}-\ell_i+n$ for $i=3,4,\ldots,c$. Assume $a_{i-\min([c/2]+1,t)}\ge b_i$ for $\min([c/2]+1,t)\le i\le t$.

The dimension conjecture for good determinantal schemes. Under these assumptions,

$$\dim W(\underline{b};\underline{a})=\lambda_c+K_3+K_4+\cdots+K_c,$$

where $K_3=\binom{h_0}{n}$,

$$K_4=\sum_{j=0}^{t+1}\binom{h_1+a_j}{n}-\sum_{i=1}^{t}\binom{h_1+b_i}{n},$$

and, for $0\le i\le c-3$,

$$K_{i+3}=\sum_{r+s=i\atop r,s\ge0}\sum_{0\le i_1<\cdots<i_r\le t+i\atop 1\le j_1\le\cdots\le j_s\le t}(-1)^{i-r}\binom{h_i+a_{i_1}+\cdots+a_{i_r}+b_{j_1}+\cdots+b_{j_s}}{n}.$$

In particular, when all entries of the matrix have the same degree, this specializes to $\dim W(\underline{0};\underline{d})=t(t+c-1)\binom{d+n}{n}-t^2-(t+c-1)^2+1$. The formula predicts the dimension of the determinantal locus and is known in some cases, including the ranges cited in the surrounding discussion, but is not established in general.