**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# Homogeneous coordinate ring

Summary

In algebraic geometry, the homogeneous coordinate ring R of an algebraic variety V given as a subvariety of projective space of a given dimension N is by definition the quotient ring
:R = K[X0, X1, X2, ..., XN] / I
where I is the homogeneous ideal defining V, K is the algebraically closed field over which V is defined, and
:K[X0, X1, X2, ..., XN]
is the polynomial ring in N + 1 variables Xi. The polynomial ring is therefore the homogeneous coordinate ring of the projective space itself, and the variables are the homogeneous coordinates, for a given choice of basis (in the vector space underlying the projective space). The choice of basis means this definition is not intrinsic, but it can be made so by using the symmetric algebra.
Formulation
Since V is assumed to be a variety, and so an irreducible algebraic set, the ideal I can be chosen to be a prime ideal, and so R is an integral domain. The same definition can be used for general homogeneous ideals, but t

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications

Loading

Related people

Loading

Related units

Loading

Related concepts

Loading

Related courses

Loading

Related lectures

Loading

Related publications

Related units

No results

No results

Related concepts

Related people

No results

No results

Related courses

Related lectures

No results

No results