Real Legendrian subvarieties are classical objects of differential geometry and classical mechanics and they have been studied since antiquity (see [Arn74], [Sła91] and references therein). However, complex Legendrian subvarieties are much more rigid and have more exceptional properties. The most remarkable case is the Legendrian subvarieties of projective space; prior to the author's research only few smooth examples of these were known (see [Bry82], [LM07]). Strong restrictions on the topology...
The Cox ring provides a coordinate system on a toric variety analogous to the homogeneous coordinate ring of projective space. Rational maps between projective spaces are described using polynomials in the coordinate ring, and we generalise this to toric varieties, providing a unified description of arbitrary rational maps between toric varieties in terms of their Cox coordinates. Introducing formal roots of polynomials is necessary even in the simplest examples.
We study semigroups of labellings associated to a graph. These generalise the Jukes-Cantor model and phylogenetic toric varieties defined in [Buczynska W., Phylogenetic toric varieties on graphs, J. Algebraic Combin., 2012, 35(3), 421–460]. Our main theorem bounds the degree of the generators of the semigroup by g + 1 when the graph has first Betti number g. Also, we provide a series of examples where the bound is sharp.
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