Complementary Modules of Weierstrass Canonical Forms

The Weierstrass curve is a pointed $(X,\infty)$ with numerical semigroup $H_X$, which normalization of the given by canonical form, $y^r + A_{1}(x) y^{r-1} A_{2}(x) y^{r-2} +\dots A_{r-1}(x) y A_{r}(x)=0$ where each $A_j$ polynomial in $x$ degree $\leq j s/r$ for certain coprime positive integers $r$ and $s$, $r$<$s$, such that generators non-gap sequence $H_X$ at $\infty$ include $s$. has projection $\varpi_r\colon X \to {\mathbb P}$, $(x,y)\mapsto x$, as covering space. Let $R_X := {\mathbf H}^0(X, {\mathcal O}_X(*\infty))$ $R_{\mathbb P} H}^0({\mathbb P}, O}_{\mathbb P}(*\infty))$ whose affine part ${\mathbb C}[x]$. In this paper, every $X$, we show explicit expression complementary module $R_X^{\mathfrak c}$ P}$-module $R_X$ an extension plane curves Kunz. naturally leads expressions holomorphic one form except $\infty$, ${\mathbf A}_{\mathbb terms $R_X$. Since compact Riemann surface, find bi-rational to also comment algebraic construction generalized Weierstrass' sigma functions surface connected data on how embedded into universal Grassmannian manifolds.