Where algebra meets geometry — explore varieties, curves, and surfaces through 28 interactive demonstrations spanning affine varieties, projective space, elliptic curves, the Nullstellensatz, blow-ups, and the 27 lines on a cubic surface.
See also: Beautiful Varieties for a gallery of the field's most stunning objects, Rings & Fields for the coordinate rings that encode every variety algebraically, Differential Geometry for the smooth structures behind complex projective varieties, and Category Theory for the functorial perspective Grothendieck made central.
From polynomials to geometry — zero sets in affine space and their coordinate rings
Lines through the origin, points at infinity, and the projective plane
The Cayley cubic, Clebsch diagonal, Barth sextic, and Kummer surface — rendered in full glory
When two curves of degrees m and n meet, count mn points — if you look in the right place
Point addition, the chord-tangent construction, and the group structure on a cubic
Rings ↔ spaces, ideals ↔ subvarieties, primes ↔ irreducibles — the Rosetta Stone of algebraic geometry
Hilbert's theorem connecting algebra and geometry — ideals correspond to varieties
Resolving singular points by replacing them with projective lines — watch singularities untangle
Every smooth cubic surface contains exactly 27 lines — find them all