A triangle is the projective plane, a square is a product of lines — the magic dictionary of toric varieties
In one corner of algebraic geometry you can answer hard questionsby looking at a polygon: corners, edges, and dots each stand for a piece of a surface.
Pick one of five polygons. Hover a corner, an edge, or the interior to see the part of the surface it stands for, and the self-intersection numbers along the edges.
3 lattice points ↔ the 3 monomials 1, x, y — the linear forms on P². The three edges are the three coordinate lines X=0, Y=0, Z=0.
Hover a corner, an edge, or the interior. Each part of the polygon is a part of the surface: corners are points, edges are spheres, and the inside is a torus.
Counting lattice points is doing geometry. The number of lattice points in the polygon is the dimension of the space of sections of the corresponding line bundle — so the dilated triangle 2Δ, with its 6 lattice points, is telling you that a conic’s equation has exactly 6 coefficients, and hence that conics themselves, defined up to scale, form a 5-dimensional family. A question about counting dots has become a question about curves.
The triangle with corners (0,0), (1,0), (0,1) is the projective plane. The unit square is P1 × P1, a product of two spheres. Chop a corner off the triangle and you have blown up a point. Count the dots inside a polygon and you have computed the dimension of a space of curves. This is the theory of toric varieties, and it is the closest thing algebraic geometry has to a picture book.
The starting point is the algebraic torus (C*)2 — pairs of nonzero complex numbers, with multiplication. It is a perfectly good algebraic variety, but not a compact one: points can run off to zero or to infinity. A toric variety is what you get by adding boundary to fix that, and a polygon is a set of instructions for which boundary to add.
Each edge of the polygon has an inward-pointing normal direction, and those directions together form the normal fan. Each fan ray tells you to add one boundary curve — a copy of P1 — that catches the points escaping in that direction. Each corner, where two rays meet, tells you where two of those boundary curves cross. The triangle has three edges and three corners, so P2 gets three boundary lines meeting pairwise in three points: exactly the three coordinate lines X = 0, Y = 0, Z = 0.
The polygon carries more information than the fan alone. The fan builds the space; the polygon also chooses an embedding of it. That is why the triangle Δ and the doubled triangle 2Δ give the same surface P2 but different pictures: Δ has 3 lattice points and embeds P2 in P2 as itself, while 2Δ has 6 and embeds it in P5 by the six quadratic monomials — the Veronese surface. Same space, different coat. The projective plane in its own right is explored in the Algebraic Geometry module’s Projective Space lesson.
Drag the cursor around the triangle and watch the fibre above it: a torus inside, a circle on an edge, a point at a corner. The one-dimensional warm-up turns a segment into a sphere.
The moment map takes the surface and flattens it onto the polygon, crushing each torus orbit to a single point. Nothing is thrown away that cannot be put back: over an interior point sits a full two-dimensional torus, over an edge the torus has lost a circle’s worth of freedom, and over a corner it is gone entirely. Slide the cursor to an edge and watch the tube deflate; slide it into a corner and it vanishes. The whole four-real-dimensional surface is encoded in a flat triangle.
The compact torus — pairs of phases, a doughnut — acts on a toric variety by rotating the coordinates. The moment map sends each point of the surface to a point of the polygon, and its fibres are exactly the orbits of that action. So the polygon is the space of orbits, flattened out.
Over an interior point the orbit is a full 2-torus: both phases can turn freely. Over an edge, one of the coordinates has hit zero, so its phase no longer means anything and the orbit is only a circle. Over a corner, two coordinates have gone, and the orbit is a single point that the torus cannot move at all. That is why the corners are called fixed points.
Add up the dimensions: 2 real dimensions of polygon plus 2 of torus fibre gives 4, and a complex surface has exactly 4 real dimensions. Nothing is lost. Atiyah, and independently Guillemin and Sternberg, proved in 1982 that the image of a moment map is always a convex polytope — which is why this picture is available at all.
Click a corner to chop it. Each chop is a blow-up: a new curve of self-intersection −1 appears, and the fan gains a ray. Chop all three corners.
The unchopped triangle. All three edges have self-intersection +1 — the three coordinate lines of the projective plane, each meeting the others once.
The self-intersection numbers are not decoration — they are forced. For a smooth toric surface the primitive normals obey ui−1 + ui+1 = −ai·ui, and that equation determines every number on the picture. Chop all three corners of the triangle and every edge becomes a (−1)-curve: six of them, on the most symmetric del Pezzo surface there is.
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