Symplectic mapping class relations from pencil pairs

We describe symplectic mapping class relations between products of positive Dehn twists along Lagrangian spheres in Weinstein $4$-manifolds, all of which are affine $\mathbb{C}$ varieties. The relations are obtained by applying classification results for Fano $3$-folds and polarized $K3$ surfaces of small genus to a general methodology – finding pencil pairs.