Imaginary Projections: Complex Versus Real Coefficients

Given a multivariate complex polynomial ${p\in\mathbb{C}[z_1,\ldots,z_n]}$, the imaginary projection $\mathcal{I}(p)$ of $p$ is defined as the projection of the variety $\mathcal{V}(p)$ onto its imaginary part. We focus on studying the imaginary projection of complex polynomials and we state explicit results for certain families of them with arbitrarily large degree or dimension. Then, we restrict to complex conic sections and give a full characterization of their imaginary projections, which generalizes a classification for the case of real conics. That is, given a bivariate complex polynomial $p\in\mathbb{C}[z_1,z_2]$ of total degree two, we describe the number and the boundedness of the components in the complement of $\mathcal{I}(p)$ as well as their boundary curves and the spectrahedral structure of the components. We further show a realizability result for strictly convex complement components which is in sharp contrast to the case of real polynomials.Comment: 24 pages; Revised versio

Real Space Sextics and their Tritangents

The intersection of a quadric and a cubic surface in 3-space is a canonical curve of genus 4. It has 120 complex tritangent planes. We present algorithms for computing real tritangents, and we study the associated discriminants. We focus on space sextics that arise from del Pezzo surfaces of degree one. Their numbers of planes that are tangent at three real points vary widely; both 0 and 120 are attained. This solves a problem suggested by Arnold Emch in 1928

Tritangents and their space sextics

Two classical results in algebraic geometry are that the branch curve of a del Pezzo surface of degree 1 can be embedded as a space sextic curve and that every space sextic curve has exactly 120 tritangents corresponding to its odd theta characteristics. In this paper we revisit both results from the computational perspective. Specifically, we give an algorithm to construct space sextic curves that arise from blowing up projective plane at eight points and provide algorithms to compute the 120 tritangents and their Steiner system of any space sextic. Furthermore, we develop efficient inverses to the aforementioned methods. We present an algorithm to either reconstruct the original eight points in the projective plane from a space sextic or certify that this is not possible. Moreover, we extend a construction of Lehavi which recovers a space sextic from its tritangents and Steiner system. All algorithms in this paper have been implemented in magma.Comment: 24 pages, 2 figure