Unexpected hypersurfaces: some examples, a few constructions

Abstract: In recent years, a novel attitude to the classical problem of identifying and classifying special linear systems in projective $n$ space, has been emerged. For a subvariety $Z$ of the projective $n$ space with defining ideal $I$, let $P_1,\dots,P_s$ be general distinct points in this space and let $m_1,\dots,m_s$ be positive integers which at least one of them is greater than one. On the subspace of those elements of degree $d$ part of the homogeneous ideal $I$ which vanish at $P_i$ with multiplicity at least $m_i$, each fat point $m_iP_i$ defines a specific number of linear relations on this subspace. For a given set of points $P_i$ with multiplicity $m_i$, $1\le i \le s$, it is expected that these linear equations to be linearly independent. If it is not the case, then one says that the variety $Z$ admits an unexpected hypersurface with respect to fat point subscheme defined by these fat points, and this linear subspace is called a special linear system on the variety $Z$. Each element of this subspace, defines a hypersurface, known as unexpected hypersurface. In this talk, we review some interesting examples which brought into the scene with this new approach and explain some existing methods to construct unexpected hypersurfaces.

algebraic geometry

Audience: researchers in the topic

Comments: https://zoom.us/join

Meeting ID: 9086116889

Passcode: 13440 $\times$ the number of lines on a cubic surface

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IPM Algebraic Geometry Seminar

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