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A Pfaffian equation on $X$ is an invertible subsheaf $L \hookrightarrow \Omega^1_X$ of the sheaf of differential 1-forms on $X$. It can thus be thought of as a global section of $\Omega^1_X \otimes L^{-1}$ or, by choosing an isomorphism $L^{-1} \simeq {\cal O}_X(E) \subset K$, as a meromorphic differential form $\omega$ on $X$. We say that the Pfaffian equation has a first integral if there is a non-empty open subset $U\subset X$ and a smooth map $f: U\rightarrow {\rm P}^1$ such that $L_U \simeq f^* \Omega^1_{{\rm P}^1}$ as subsheaves of $\Omega^1_X$; that is, if $L$ ‘comes from’ a rational map to a curve. In this case, $f$, viewed as a rational function, is called a first integral. Note that as rational differential forms, ${\rm d} f \wedge \omega = 0$.

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Pfaffian equations and the Cartier operator

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Pfaffian equations and the Cartier operator