4 problems

Positivity and monotonicity conjecture. The Welschinger invariants $W_{D,m}$ are positive if $m<[(c_1(\Sigma)\cdot D-1)/2]$, non-negative if $m=[(c_1(\Sigma)\cdot D-1)/2]$, and sat…

Sharpness conjecture. If $X$ is $\mathbb P^2$, $\mathbb P^1\times\mathbb P^1$, or a Hirzebruch surface, then

Let $X$ be a smooth complete toric surface, let $\mathcal{L}$ be a line bundle in the setting of the monodromy map, and let $C$ be a curve whose nodes form the set on which monodro…

Let $X$ be a smooth toric surface and let $\mathcal L$ be an ample line bundle on $X$ whose generic fiber is not hyperelliptic. Assume $r>1$ or $g(\mathcal L)\geq 5$, and let…