9 problems

Let $G$ be a reductive group with maximal torus $T$, and let $Y$ be a $T$-stable variety in $G/B$. At a $T$-fixed point, consider the tangent space and the corresponding Peterson t…

Second additional family conjecture. The ideal $I$ has colength $\binom{k+2}{3}+k+3$ and has maximum tangent-space dimension; moreover,

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 $N\geq 3$. 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…

Tangent-space parity conjecture. For every point $P\in \operatorname{Hilb}^n(\mathbb{A}^3)$,

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 $r$,…

Let $\mathbf{k}$ be a field of characteristic zero, let $d={r+2\choose3}$, and let $[I]\in\mathrm{Hilb}^{d}\mathbf{A}^3$ be Borel-fixed. Write $\mathfrak{m}=(x_1,x_2,x_3)$, and let…

Scalar multiplication conjecture. For every $r\in\mathbb{R}$, the induced map is scalar multiplication by $r$:

Let $\Omega$ be an irreducible analytic set in a space of convergent power series, and let $f\in\Omega$. Define the algebraic tangent space by … and the geometric tangent space by…