Special geometry of quartic curves

David Lindemann

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arXiv · 2206.12524

Special geometry of quartic curves

David Lindemann

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Abstract

We classify maximal quartic generalised projective special real curves up to equivalence. A maximal quartic generalised projective special real curve consists of connected components of the intersection of the hyperbolic points of a quartic homogeneous real polynomial $h:\mathbb{R}^2\to\mathbb{R}$ and its level set $\{h=1\}$. Two such curves are called equivalent if they are related by a linear coordinate transformation. As an application of our results we prove that quartic generalised projective special real manifolds that are homogeneous spaces have non-regular boundary behaviour, meaning that the differential of each of these spaces' defining polynomials vanishes identically on a ray in the boundary of the cone spanned by the corresponding manifold. Lastly we describe the asymptotic behaviour of each curve.

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David Lindemann. 2022-06-25. Special geometry of quartic curves. https://arxiv.org/abs/2206.12524

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