311 research outputs found
Quartic Curves and Their Bitangents
A smooth quartic curve in the complex projective plane has 36 inequivalent
representations as a symmetric determinant of linear forms and 63
representations as a sum of three squares. These correspond to Cayley octads
and Steiner complexes respectively. We present exact algorithms for computing
these objects from the 28 bitangents. This expresses Vinnikov quartics as
spectrahedra and positive quartics as Gram matrices. We explore the geometry of
Gram spectrahedra and we find equations for the variety of Cayley octads.
Interwoven is an exposition of much of the 19th century theory of plane
quartics.Comment: 26 pages, 3 figures, added references, fixed theorems 4.3 and 7.8,
other minor change
Plane quartics: the universal matrix of bitangents
Aronhold's classical result states that a plane quartic can be recovered by the configuration of any Aronhold systems of bitangents, i.e. special -tuples of bitangents such that the six points at which any sub-triple of bitangents touches the quartic do not lie on the same conic in the projective plane. Lehavi (cf. \cite{lh}) proved that a smooth plane quartic can be explicitly reconstructed from its bitangents; this result improved Aronhold's method of recovering the curve. In a 2011 paper \cite{PSV} Plaumann, Sturmfels and Vinzant introduced an
symmetric matrix that parametrizes the bitangents of a nonsingular plane quartic. The starting point of their construction is
Hesse's result for which every smooth quartic curve has exactly equivalence classes of
linear symmetric determinantal representations.
In this paper we tackle the inverse problem, i.e. the construction of the bitangent matrix starting from the 28 bitangents of the plane quartic, and we provide a Sage script intended for computing the bitangent matrix of a given curve
Bitangents of non-smooth tropical quartics
We study bitangents of non-smooth tropical plane quartics. Our main result is
that with appropriate multiplicities, every such curve has 7 equivalence
classes of bitangent lines. Moreover, the multiplicity of bitangent lines
varies continuously in families of tropical plane curves.Comment: 8 pages, 3 figures. Initiated at the Fields Undergraduate Summer
Research Progra
Computing Linear Matrix Representations of Helton-Vinnikov Curves
Helton and Vinnikov showed that every rigidly convex curve in the real plane
bounds a spectrahedron. This leads to the computational problem of explicitly
producing a symmetric (positive definite) linear determinantal representation
for a given curve. We study three approaches to this problem: an algebraic
approach via solving polynomial equations, a geometric approach via contact
curves, and an analytic approach via theta functions. These are explained,
compared, and tested experimentally for low degree instances.Comment: 19 pages, 3 figures, minor revisions; Mathematical Methods in
Systems, Optimization and Control, Birkhauser, Base
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