Veronese embeddings, arc schemes and global Demazure modules

Abstract: Veronese curve of degree d (also known as rational normal curve) can be realized as an embedding of the complex projective line into a d-dimensional projective space. The equations cutting out the image of this embedding can be written down explicitly and the homogeneous coordinate ring has an explicit description in terms of representations of the complex Lie algebra sl(2). To pass to the corresponding arc scheme, one replaces the field of complex numbers with the ring of formal Taylor series in one variable. We describe the reduced ideal of the arc scheme and the homogeneous coordinate ring in terms of representation theory of the current algebra of sl(2). The whole picture generalizes to the case of an arbitrary simple Lie algebra. The analogues of the rational normal curves are the Veronese embeddings of the flag varieties for the corresponding Lie group. We identify the homogeneous coordinate ring of the reduced arc scheme of the Veronese embedding with the direct sum of the global Demazure modules of the current algebra (the higher level analogues of the global Weyl modules). Joint work with Ilya Dumanski. Geometric flows of $G_2$ and Spin(7)-structures

Mathematics

Audience: researchers in the discipline

CRM-Regional Conference in Lie Theory

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