A gallery of the most beautiful objects in algebraic geometry — 24 interactive demonstrations showcasing world-record singular surfaces, tropical curves and their amoebas, the 3264 conics, the Hesse configuration, ADE singularities, elliptic curves as doughnuts, polygons that are secretly spaces, and the wild ovals of Hilbert’s 16th problem. One masterpiece per room; no prerequisites, just wonder.
See also: Algebraic Geometry for the theory behind these objects — varieties, projective space, and the algebra–geometry dictionary, Schemes for the modern foundations, and Lie Groups for the Dynkin diagrams that reappear in the ADE singularities.
Togliatti, Labs, Endraß, Barth, Sarti — the surfaces holding the world records for singular points
Take logarithms of a curve and watch it cast an amoeba-shaped shadow that crystallizes into piecewise-linear bones
How many conics are tangent to five given conics? Steiner guessed 7776. The truth is stranger — and all of them can be real
Nine inflection points on twelve lines — a configuration so symmetric it cannot be drawn with real straight lines
A sculptural gallery of the du Val singularities — and the Dynkin diagrams hiding inside each one
A lattice in the complex plane rolls up into a torus, and the torus is secretly a cubic curve
A triangle is the projective plane, a square is a product of lines — the magic dictionary of toric varieties
How many ovals can a plane curve have, and how can they nest? A zoo of wild curves ending at an open problem