The partially strict unimodal criterion for 2-admissible I-Gorenstein sequences

Let $A=k[\

x,y ]\

/J$be an AG$k$-algebra. Let$H=(h_0,h_1,\ldots,h_u)$be a$2$-admissible sequence, where a sequence is **partially strict unimodal** if$h_0<h_1<\cdots<h_i\geq h_{i+1}\geq\cdots\geq h_u$for some$0\leq i\leq u$. Suppose that

$$|h_i-h_{i+1}|\leq h_0$$

for all $i$. The partially strict unimodal criterion. Then $H$ is an $I$-Gorenstein sequence. This proposes a sufficient condition for realizing such sequences as Hilbert functions of associated graded rings of ideals in AG algebras; the paper notes that the condition is supported by examples and low-degree investigation, while the general assertion remains open.