6 problems

Let $G$ be a graph formed by gluing $k$ copies of the four-cycle $C_{4}$ together along one edge. Write $b_{i,j}(\overline{G})$ for the graded Betti numbers of its graph curve. Bet…

Let $G$ be a graph and let $\overline{G}\subset \mathbb{P}^{n}$ be its graph curve. Write $b_{i,j}(\overline{G})$ for the graded Betti numbers, and call an edge of $G$ a bridge if…

Let $G$ be a graph with $d$ vertices, genus $g$, and girth $b3$. Let $\overline{G}$ be the corresponding graph curve, with Betti numbers $b_{i,j}(\overline{G})$. Burnham–Rosen–Sidm…

Let $C_G\subset\mathbb{P}^{g+r}$ be a graph curve of arithmetic genus $p_a$ and degree $d=2g+r$. A $(k+2)$-secant $\mathbb{P}^k$ is a projective $k$-plane meeting $C_G$ in a subsch…

Let $G$ be a graph on $d$ vertices, embedded as in Theorem 1.3. Let $n$ denote the girth of $G$, and suppose that $d=2g+1+p$ and $n-2\leq p$. Minimal-cycle Betti-number conjecture.…

Let $C_G$ be a graph curve embedded under the hypotheses of Theorem 1.3, and let $k$th secant variety denote the union of the linear spans of length-$k+1$ subschemes of $C_G$. Seca…