Geometry of some moduli of bundles over a very general sextic surface for small second Chern classes and Mestrano-Simpson conjecture

Let $S \subset \mathbb P^3$ be a very general sextic surface over complex numbers. $\mathcal{M}(H, c_2)$ the moduli space of rank $2$ stable bundles on $S$ with fixed first Chern class $H$ and second $c_2$. In this article we study configuration points certain reduced zero dimensional subschemes satisfying Cayley-Bacharach property, which leads to existence non-trivial sections memeber for small Using will make an attempt prove Mestrano-Simpson conjecture number irreducible components 11)$ partially. We also show that is $c_2 \le 10$ .