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Recovering an algebraic curve using its projections from different points. Applications to static and dynamic computational vision

Jeremy Yirmeyahu Kaminski, Michael Fryers, Mina Teicher (2005)

Journal of the European Mathematical Society

We study some geometric configurations related to projections of an irreducible algebraic curve embedded in ℂ ℙ 3 onto embedded projective planes. These configurations are motivated by applications to static and dynamic computational vision. More precisely, we study how an irreducible closed algebraic curve X embedded in ℂ ℙ 3 , of degree d and genus g , can be recovered using its projections from points onto embedded projective planes. The embeddings are unknown. The only input is the defining equation of...

Sagbi bases of Cox–Nagata rings

Bernd Sturmfels, Zhiqiang Xu (2010)

Journal of the European Mathematical Society

We degenerate Cox–Nagata rings to toric algebras by means of sagbi bases induced by configurations over the rational function field. For del Pezzo surfaces, this degeneration implies the Batyrev–Popov conjecture that these rings are presented by ideals of quadrics. For the blow-up of projective n -space at n + 3 points, sagbi bases of Cox–Nagata rings establish a link between the Verlinde formula and phylogenetic algebraic geometry, and we use this to answer questions due to D’Cruz–Iarrobino and Buczyńska–Wiśniewski....

Subvarieties of the Hyperelliptic Moduli Determined by Group Actions

Shaska, T. (2006)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 14Q05, 14Q15, 14R20, 14D22.Let Hg be the moduli space of genus g hyperelliptic curves. In this note, we study the locus Hg (G,σ) in Hg of curves admitting a G-action of given ramification type σ and inclusions between such loci. For each genus we determine the list of all possible groups, the inclusions among the loci, and the corresponding equations of the generic curve in Hg (G, σ). The proof of the results is based solely on representations of finite subgroups...

Taylorian points of an algebraic curve and bivariate Hermite interpolation

Len Bos, Jean-Paul Calvi (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We introduce and study the notion of Taylorian points of algebraic curves in ℂ 2 , which enables us to define intrinsic Taylor interpolation polynomials on curves. These polynomials in turn lead to the construction of a well-behaved Hermitian scheme on curves, of which we give several examples. We show that such Hermitian schemes can be collected to obtain Hermitian bivariate polynomial interpolation schemes.

The degree of the secant variety and the join of monomial curves.

Kristian Ranestad (2006)

Collectanea Mathematica

A monomial curve is a curve parametrized by monomials. The degree of the secant variety of a monomial curve is given in terms of the sequence of exponents of the monomials defining the curve. Likewise, the degree of the join of two monomial curves is given in terms of the two sequences of exponents.

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