31 problems

Self-intersection conjecture. One has

Let $X$ be an elliptic ruled surface with $e=-1$, and let $L\in \operatorname{Pic}X$ be a line bundle whose numerical class is $aC_0+bf$. Property $N_p$ is the syzygy condition on…

Extension conjecture. Theorem also holds for sporadic parabolically polystable ruled surfaces.

Ruled-surface characterization. There exists a counterexample to the Kawamata–Viehweg vanishing theorem on some geometrically ruled surface over $C$ if and only if $C$ is a Tango c…

Parabolic stability necessity conjecture. If $\widehat M$ admits a scalar-flat Kähler metric, then $E$ must be parabolically stable. This is a necessary-condition conjecture for th…

LeBrun–Simanca conjecture. The Kähler class $[\omega]$ contains a scalar-flat Kähler representative if and only if the parabolic bundle $E$ is stable. This conjectures that the qua…

Let $X$ be a ruled surface over a smooth projective curve $C$ of genus $g$, and let $L\in \operatorname{Pic}X$ be a line bundle in the numerical class $aC_0+bf$. Here $e<0$ is the…

Let $X$ be a ruled surface over a smooth projective curve $C$ of genus $g$, and let $L\in \operatorname{Pic}X$ be a line bundle in the numerical class $aC_0+bf$, with $a\geq 1$; le…

Let $X$ be a ruled surface over a smooth projective curve $C$ of genus $g$, and let $L\in \operatorname{Pic}X$ be a line bundle in the numerical class $aC_0+bf$, with $a\geq 1$. He…

Smoothness conjecture. Every component of the Hilbert scheme of $X$ is smooth of the expected dimension.

Let $M$ be a blown-up ruled surface of genus at least $2$, and let $[\omega]$ be an admissible Kähler class, meaning that $c_1\cup[\omega]=0$. Associate to $[\omega]$ its correspon…

Let $M$ be a ruled compact complex surface, and let $\widetilde{M}$ be its blow-up at sufficiently many points. Blow-up conjecture. The surface $\widetilde{M}$ admits a scalar-flat…

Let … and suppose that $P$, $Q$, and $R$ are linear combinations of polynomials $P_1$, $Q_1$, and $R_1$ satisfying Conjecture 1, with the combinations coupled by a non-degenerate m…

Let … where $W_c(s)$ is the transfer function of a linear subsystem and the polynomial factor is linear in $k=(k_1,k_2,k_3)$. Ruled-boundary conjecture. If each of the polynomials…

Let $X$) be the minimal ruled surface considered in the paper, with distinguished sections $S_0$ and $S_\infty$, and let $\omega$ be a conical higher cscK metric on $X$ with cone a…

Let $\mathbb{F}_e$ be the $e$-th Hirzebruch surface, with $C_e$ denoting its negative section. Let $P=\{p_1,\dots,p_r\}\subset \mathbb{F}_e$ be $r$ points in very general position,…

Let $x(u_1,u_2)$ be the special ruled surface described by a directrix $\Gamma(u_2)$ and unit ruling direction $\Phi(u_2)$, with the input crease-length distribution encoded by…

Select one row and one column of crease lengths as the input crease lengths in a quad-mesh rigid origami generated by repetitive stitching, and suppose the input crease-length dist…

Let $u_1,u_2\in\mathbb{R}$ and let $x(u_1,u_2)\in\mathbb{R}^3$ be given by … where … with $a>0$, $b\in\mathbb{R}$, and … with … Here $f(u_2)$ is a known input crease-length distrib…

A quad-mesh rigid origami is a structure of planar, rigid, zero-thickness quadrilateral panels connected by rotational hinges in a grid-like configuration. Consider the quad-mesh r…

Sols and Sols' conjecture. If

Let $F_1$ and $F_2$ denote the two ruling classes on $\mathbb{P}^1\times\mathbb{P}^1$. A vector bundle $\mathcal{V}$ has natural cohomology if $\mathcal{V}(aF_1+bF_2)$ has at most…

Pingali's conjecture. There is a maximum value of $m$ beyond which no solution to this problem exists; equivalently, higher extremal Kähler metrics need not exist in the classes…

Stability conjecture. The reduced symplectic cone should admit a chamber partition such that the homotopy type of the symplectomorphism group is invariant within each chamber.

Let $M_g=\Sigma_g\times S^2$ be a minimal topologically trivial ruled surface equipped with a symplectic form $\omega$. A symplectomorphism of $(M_g,\omega)$ is symplectically isot…