3 problems

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The converse conjecture for ACM point sets and the condition $N_1(X) \in \mathcal D(X)$
Converse conjecture. Then
Conjectureopenarithmetically-cohen-macaulay finite-point-setsarXiv posted 2 days ago
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The inclusion-property conjecture for finite point sets in multiprojective spaces
Inclusion-property conjecture. $X$ is arithmetically Cohen–Macaulay (ACM).
Conjectureopenarithmetically-cohen-macaulay finite-point-sets multiprojective-spacesarXiv posted 2 days ago
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Classification and arithmetically Cohen–Macaulayness of line sets of type $(d-1,d)$
Classification and ACM conjecture. Every finite set of lines in ${\mathbf P}^3$ has type $(d-1,d)$ if and only if it is coplanar or is the intersection of a finite set of hyperplan…
Conjectureopenarithmetically-cohen-macaulay line-configurations projective-geometryarXiv posted 2 days ago

The converse conjecture for ACM point sets and the condition N1(X)∈D(X)N_1(X) \in \mathcal D(X)N1​(X)∈D(X)