16 problems

Sharp logarithmic norm conjecture. The quantity $(\log \delta^{-1})^{1/p}$ is the sharp $L^p$ operator norm of $\mathcal{N}_{\delta}^{\gamma}$ for $p\in[2,\infty)$; that is, the lo…

Kleppe–Ellia conjecture. The family $W$ is an irreducible component of $(H(d,g)^{sc})_{\operatorname{red}}$, and $H(d,g)^{sc}$ is generically non-reduced along $W$.

Let $d,m$ be positive integers in Range B, namely … Let $G(d,m)$ denote the maximal genus of a smooth connected curve of degree $d$ in $\mathbb{P}^3$ not contained in a surface of…

Let $W$ be a $3$-maximal family of smooth connected, linearly normal space curves of degree $d>9$ and genus $g$, whose general member $C$ is contained in a smooth cubic surface. Wr…

Let $C\subset\mathbb P^3$ be the union of $r$ general lines. The strong Lefschetz property conjecture. Then $M(C)$ has SLP. Here $M(C)$ denotes the Hartshorne–Rao module of $C$, an…

Constant-curvature W-curve conjecture. Space curves with both $\kappa_i$ constant may be related to $W$-curves.

Square-like quadrilateral conjecture. There is a square-like quadrilateral inscribed in any embedding of $S^1$ in $\mathbb{R}^n$.

Let $C\subset\mathbb{P}^r$ be a general Brill–Noether curve of degree $d$ and genus $g\geq 2$, where $r\geq 3$. Its normal bundle is denoted by $N_{C/\mathbb{P}^r}$. Brill–Noether…

Fix positive integers $d,m$. In Range A, the relevant curves are smooth curves $C\subset\mathbb{P}^3$ of degree $d$ with $h^0(\mathcal{I}_C(m-1))=0$; in Range B, take … and in Rang…

General weighted-homogeneous polynomial ideal conjecture. The ideal $I$ contains no polynomial of standard degree $3m-3$. If true, the construction in the paper would imply the rel…

Let $d \geq 5$ be an integer congruent to $2$ modulo $3$. A primitive $d$-line is a primitive locally Cohen–Macaulay $d$-fold structure supported on a line $L\subset\mathbb{P}^3$;…

A primitive multiple line is a locally Cohen–Macaulay multiple structure supported on a line in $\mathbb{P}^3$; let $\mathcal{C}$ be the large family of primitive multiple lines co…

Maximal-rank conjecture. There exists a constant $K$ such that, for all natural numbers $d,g$ satisfying

Let $L_5$ be the closed $C^1$-smooth space curve formed from the four congruent portions described in the paper, and let $L$ and $w$ denote its length and width. Zalgaller's $L_5$…

Let $Z$ be the space curve constructed in the paper, and let $w(Z)$ denote its width. Zalgaller's conjecture. The curve $Z$ has width … The paper's computation gives an upper bound…

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be cubic rational helical space curves with proportionality constants $\mu_1$ and $\mu_2$, respectively. The curves have proportionality con…