Let $S$ be a Weierstrass model of an elliptic surface (for me it works better to understand it as an elliptic fibration), that is a map $\pi : S \to C$ where $C$ is a compact Riemann surface.
Disclaimer: Physicist here.
Intuitively I think of this as a compact Riemann surface (e.g. $\mathbb{CP}^1$) where over each point $z$ we assign an elliptic curve in the Weierstass form (please let me know if this is wrong). In physics such a system defines the low energy dynamics of the $\mathcal{N}=2$ super Yang-Mills theory on a 4-manifold. The elliptic curves in the Weierstrass model are cubics of the form $$ y^2 = x^3 - g_2x - g_3 $$ where $g_2= 60 G_4(\tau)$ and $g_3=140 G_6(\tau)$ with $G_i(\tau)$ the $i$-th Eisenstein series and $\tau$ the complex structure of the elliptic curve.
Then, let $O_S$ denote the sheaf of rational functions on $S$. We can construct the pushforward sheaf on $C$ as $\pi_{*}O_S$. This is the projection of the sheaf $O_S$ from $S$ to the base $C$. Then one can view the invariants $g_2$ and $g_3$ as sections of a specific line bundle that I describe below.
This is taken from Friedman's and Morgan's book "Smooth Four-Manifolds and Complex Surfaces".
In page 57 they define the sheaf (of rank 1) $L$ as the dual of the sheaf $R^1\pi_{*}O_S$ which is a line bundle on $L$. Actually $g_2 \in H^0(C, L^{\otimes 4})$ and $g_3 \in H^0(C, L^{\otimes 6})$
Questions:(keep in mind I am a physicist and I will be happy with somewhat hand wavy answers)
- What is the difference between $\pi_{*}O_S$ and $R^1\pi_{*}O_S$? I understand that the latter is acted by the higher right derived functor and probably makes it a sheaf of sheaf cohomology groups but I struggle to understand what exactly this sheaf represents in $C$.
- Why $g_2 \in H^0(C, L^{\otimes 4})$ and $g_3 \in H^0(C, L^{\otimes 6})$?
- Is there a practical way (say to pick a coordinate chart, etc) and see that $L$ is of rank 1, see what its sections are and see how the $g_2$ and $g_3$ arise?
- Since $g_2$ and $g_3$ are multiples of the Eisenstein series, is it ok to understand the latter as sections of this bundle (or an isomorphic one) as well?
