9 problems

Let $L(\Delta)$ be the line bundle associated with a lattice polygon $\Delta$, and suppose it satisfies the relevant positivity condition, namely $\delta$-very ampleness. Let the t…

Let $\Delta$ be a lattice polygon, let $g$ be its number of interior lattice points, and let $S$ be a set of $n+1-g$ points in the dense torus of the associated polarized toric sur…

Let ${\mathscr C}\to U$ be a family of integral curves on a smooth surface, and let $N_r({\mathscr C})$ be the polynomials defined by … Assume that $U\cong {\mathbb P}^\delta$ and…

Fix $\delta\ge1$, and write the refined plane Severi degree as … Let $N_\delta(d;y)$ be the refined node polynomial, with coefficients $p^\delta_i(d)$ defined analogously. Coeffici…

Let $S$ be a smooth projective surface and let $L$ be a $\delta$-very ample line bundle. Let $\widetilde N^{(S,L),\delta}(y)$ be the refined universal invariant. Positivity conject…

Let $N_\delta(d,m;y)$ be the polynomial from the weighted-projective refined node polynomial conjecture, let $\widetilde N^{(\Sigma_m,dH),\delta}(y)$ be the refined invariant on th…

Let $S$ be a classical toric del Pezzo surface and let $L$ be a line bundle. Assume that the loci in $|L|$ of non-reduced curves and of curves having a $(-1)$-curve as a component…

Let $\Delta$ be a convex lattice polygon, let $S=X(\Delta)$ be the associated smooth toric surface, and let $L=L(\Delta)$ be a $\delta$-very ample line bundle. Let…

For a projective surface $S$ and line bundle $L$, let $N^{(S,L),\delta}(y)$ denote the refined Severi degree and let $\widetilde N^{(S,L),\delta}(y)$ denote the refined universal i…