20 problems

Let $m$ be a positive integer, and consider the triangular array of polynomials $H_{ij}(m)$ with interior entries, together with the least positive integer $\varkappa$ for which th…

Let $m$ be a positive integer. For $0\leq i\leq m-1$ and $0\leq j\leq m-i$, let $H_{ij}(m)$ and $D(m,k)$ be the polynomials defined in the paper, and let…

Let $c_{ijk}$ be the coefficient attached to the stabilized component $C_{ijk}$. Coefficient-one uniqueness conjecture. The components described in the preceding construction are t…

Let $\tau:P^1\times P^n\to P^{2n+1}$ be the direct-image construction, and let $X_{\mathrm{odd}}(i-1,1)_1$ and $X_{\mathrm{odd}}(i-1,1)_2$ be the two components described in the pr…

Let $H_{jk}$ be the indicated coefficients in the characteristic-polynomial construction, and let $X(\alpha,i)$ be the corresponding projective variety. Zero-locus conjecture. The…

Let $X_{\mathrm{odd}}(\alpha,i)$ be the projective scheme in $P^1(\mathbb C)\times P^\alpha(\mathbb C)$ defined by the first $i$ coefficients of the odd characteristic polynomial.…

Let $C_{ijk}$ be a component of $X(m,i)$ and let $\goth L_{e;m}(C_{ijk})$ be its transformed component under the construction discussed in Section 8. Write its parameters as…

Let $C_{ijk}(m)$ be defined by … where $d=m-(i+j)$. Persistence conjecture. Under this definition, all assertions of Main Conjecture 2.4 hold. The statement is explicitly condition…

Let $C_{ijk}(m)$ be the component corresponding to $C_{ijk}\in\operatorname{Irr}(i,j)$, let $\bar C_{ijk}\subset X(i+j,i)$ be its minimal component, and let $m\ge i+j$ with…

Let $F$ be a forest whose trees are $T_1,\dots,T_j$, with $i_\alpha$ the corresponding parameters, and for each ramification node $u$ let $q_1(u)$ and $q_2(u)$ count the nodes in t…

Let $F$ be a forest with trees $T_1,\dots,T_j$, let $d_\alpha$ be the associated depth parameters, let $\gamma_\alpha$ be the depths of the minimal contractions, and let…

Let $T$ be a rooted binary tree, let $\varphi(T,w)$ be the associated parametrization, and let $\gamma$ be the depth of the minimal contraction of $T$. Minimal-contraction degree c…

Let $\bar{\goth F}_{ij}$ be the quotient of pairs $(F,w)$ by the equivalence relation generated by $\operatorname{Aut}(F)$ and $\goth G$, and let $\varphi_{ij}$ be the induced map…

Let $F$ be a forest with weight function $w$, and let $\varphi(F,w)$ be the associated map to $X(m,i)$. Dimension conjecture. For every $F,w$, … Since the source of the parametriza…

Let $C_{ijk}(m)$ denote the irreducible components of $X(m,i)$ indexed by the stabilized component data. Rationality conjecture. All varieties $C_{ijk}(m)$ are rational. This is on…

Let $X(m,i)$ be the projective scheme defined by the first $i$ characteristic-polynomial coefficients, let $\operatorname{Irr}(X(m,i))$ be its set of irreducible components, and le…

Let $\zeta_d=\exp(2\pi\sqrt{-1}/2^d)$, and let $X(m,i)$ be the projective scheme defined by the first $i$ polynomials $D(m,j)$. Field-of-definition conjecture. All irreducible comp…

Complete-intersection conjecture. $X(m,i)$ is a complete intersection and hence has codimension $i$ in $P^m$. Thus all its irreducible components have the same codimension. The cla…

Let $T$ be a rooted binary tree and $F$ a forest equipped with a weight function $w$ satisfying the stated conditions. The construction associates to each pair $(F,w)$ an irreducib…

Let $H_{ij,n}(m)$ and $D(m,i)=H_{i0,0}(m)$ be the polynomials defined by the characteristic-polynomial coefficients above. For $q=2$, let $\langle D(m,0),\dots,D(m,i)\rangle$ denot…