Cayley cones ruled by $2$-planes: desingularization and implications of the twistor fibration

Cayley cones in the octonions $\Oc$ that are ruled byoriented 2-planes are equivalent to pseudoholomorphiccurves in the Grassmannian of oriented $2$-planes $\gro$.The well known twistor fibration $\gro \to \s{6}$ is usedto prove the existence of immersed higher-genuspseudoholomorphic curves in $\gro$. Equivalently, thisproduces Cayley cones whose links are $\s{1}$-bundles overgenus-$g$ Riemann surfaces. When the degree of an immersedpseudoholomorphic curve is large enough, the corresponding$2$-ruled Cayley cone is the asymptotic cone of anon-conical $2$-ruled Cayley $4$-fold.

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Published 1 January 2008