Refinement of the Infinitesimal Variation of Hodge Structure: the case of canonical curves

Let $C$ be a smooth complex projective curve with canonical divisor $K_C$ very ample. We explore the relation between the cup-product $$H^1({\mathrm{\Theta }}_C)⟶(H^0({\mathcal{O}}_C(K_C){)}^{\ast }\otimes H^1({\mathcal{O}}_C)$$ where ${\mathrm{\Theta }}_C={\mathcal{O}}_C(-K_C)$ is the holomorphic tangent bundle of $C$, and the geometry of the canonical embedding of $C$. The cup-product, following Griffiths, stratifies $\mathbb{P}(H^1({\mathrm{\Theta }}_C))$ by the subvarieties ${\mathrm{\Sigma }}_r$, according to the rank $r$ of $\xi \in H^1({\mathrm{\Theta }}_C)$ viewed as the linear map $$\xi :H^0({\mathcal{O}}_C(K_C))⟶H^1({\mathcal{O}}_C)$$ or, equivalently, by the dimension of the kernel of $\xi$ $$W_{\xi }=ker(\xi )\mathrm{.}$$ The refinement consists of the filtration of $W_{\xi }^{\bullet }([\phi ])$ of $W_{\xi }$, varying with $[\phi ]\in \mathbb{P}(W_{\xi })$. This filtration has geometric meaning: 1) it is related to special divisors on $C$, 2) it `counts’ certain rational normal curves in the canonical embedding of $C$. As an illustration, the results about the strata ${\mathrm{\Sigma }}_0$ and ${\mathrm{\Sigma }}_1$ are recovered and as corollaries one obtains the classical theorems of Max Noether on projective normality of the canonical embedding and Babbage-Enriques-Petri about the canonical curve being cut out by quadrics. The refinement brings out new aspects: quiver representations, Fano toric varieties with a distinguished anti-canonical divisor, dimer models. The quiver emerges from the construction and properties of the refinement; the Fano variety arises from the graph underlying the quiver and related to the Higgs structures. The graph underlying the refinement becomes an important part of the theory: it connects to topics such as the Topological Quantum field theory, moduli of elliptic curves with marked points, modular curves, higher categorical structures.