Eight-page visual primer

Algebraic varieties

See how polynomial equations become spaces, why singularities matter, and how coordinate rings let geometry and algebra answer one another.

A4 PDF, 8 pages, vector diagrams

Equations become spacesRead V(I) as the common zero set of a polynomial system.

Zoom in locallyUse gradients and the Jacobian to distinguish smooth and singular points.

Turn shapes into ringsEncode polynomial functions in the coordinate ring k[V].

Add infinityHomogenize equations and see why projective closure improves geometry.

Interactive companion

Watch smoothness fail

Vary the coefficients in y²=x³+ax+b. A curve becomes singular exactly when its cubic acquires a repeated root.

Eight concise pages covering zero sets, ideals, singularities, coordinate rings, maps, and projective closure.