4 problems

Let $k\geq 1$ and let $n=\binom{k+2}{3}+i$, where $i\in\{0,1,2,3,4,k+1,2k+1,\binom{k+3}{3}-\binom{k+2}{3}-1\}$ and $n<\binom{k+3}{3}$. Partial-classification conjecture. For each s…

Let $N\geq 1$ and let $\mathfrak{m}=(x_1,\ldots,x_N)\subset\mathbb{C}[x_1,\ldots,x_N]$. Briançon–Iarrobino conjecture. The ideal $\mathfrak{m}^k$ has the maximum dimension tangent…

Let $N\geq 3$. Let $I$ be a zero-dimensional Borel-fixed ideal in $\mathbb{C}[x_1,\ldots,x_N]$ whose convex hull is spanned by its monomial generators. Suppose that … where…

Let $\mathbf{k}$ be a field, let $S=\mathbf{k}[x_1,\ldots,x_n]$, and for $d\in\mathbf{N}$ let $E(d)\subseteq S$ be the unique lexsegment ideal such that … and, for some $r$,…