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$thm:geom$ Suppose that $C$ is a plane algebraic curve of degree $d\ge 2$. Let $p_1\ldots,p_d\in C$ be general points. Then there exists $x\in C$ such that $x+x:C\to {{\mathbb P}}^2$ is a birational map. Moreover, for every such $x$ the map $x+x:C\to {{\mathbb P}}^2$ is $K$-negative if and only if $x$ is a general point on $C$ and a general point on the tangent line to $C$ at $p_1\ldots,p_d$.

In Theorem $thm:thm1$ and Theorem $thm:thm2$ we gave theorems analogous to Theorem $thm:geom$ for plane curves of degree $d\ge 3$. To prove these theorems we used some variant of the projection formula for $r$-forms on the symmetric product of a curve. This is the projection formula ($ff$). However, to prove the projection formula ($ff$), the form is required to be symmetric. When we tried to apply this projection formula for forms of valency $3$ on curves, we ran into difficulties which do not occur when $d\ge 3$.

– To prove the projection formula ($ff$), we will need to know that the multiplication by $r$ map $\sigma_r:S^{r+1}C\to S^{r+2}C$ is injective. This requires that the twisted form of $\omega_{S^{r+2}C}$ is non-degenerate. This is a highly non-trivial inequality on invariants. In §$symmetric$ we will give a geometric description of this condition, under which we will also be able to prove it for $d\ge 3$.

– Consider the case $r=1$. Here