Introduction: how to use JavaView

You need the following

I expect that you know it, but just in case let me give this instruction: If you use full version of JavaView click right bottom of your mouse on each picture to see the menu, choose "Control Panel"; you may need to tick "View" options for seeing edges vertices and possibly to choose option "Transparency" which will let you look into the structure of each division. Next use left bottom to rotate the picture.

The source of basic pictures is provided in (hand-made, i.e. without any junk) obj file. The syntax for this class is very simple, see here.

Here is the image of a 3 dimensional simplex divided by three hyperplanes, see source file and the result which is on the left.Click the right bottom of your mouse (while over this picture) and choose option "Control Panel". Next tick "View" options: choose "Vertex", "Edge" and "Transparency", you may play with colors. More complicated pictures are obtained by merging simple pictures, with possible adjustments which are then saved in jvx code which is not suitable for viewing but provides a nicer picture: see the one on the right which illustrates Mori chamber decomposition of the movable cone coming from a resolution of a quotient symplectic singularity C^4/(Z_3)^2xZ_2 (where x stands for the semisimple product). How many symmetries does the picture have?

Example 1: convex functions on a complete fan

In Z^3 with a basis e_1, e_2, e_3 and e_0 = --(e_1 + e_2 + e_3) we shall consider fans which have rays spanned by vectors +/-- e_i.

First we look at those fans which admit strictly convex functions. Given a strictly convex function h and a > 0 we can take a convex polyhedron P_a = {v: h(v)>= -a} which, after possibly adding to h a linear function, becomes a compact polytope which contains the origin and whose faces determine the cones in the fan in the question.

We start with a polytope associated to the fan of P^3 blown up at four points: it looks like a stuffed simplex (it ate too much) which just means that letting one of its popped-up faces go down yields a simpler polytope containg the origin - and this is exactly the blow down. Rotate the picture and think about possible maps of this fan to other complete fans.

Next we modify the convex function. The result will either admit two divisorial contractions (on the left) or it will be a P^1 bundle over P^2 blown-up in 3 points (on the right hand side).

Finally, we present a fan which does not admit a strictly convex function. First we take a fan whose cones are spanned on the faces of the cube (blue faces). Next we divide the quadratic cones into simplcial cones by red faces. Two opposite faces of the cube are not divided to make the picture more transparent, you can divide them any way you like.

Example 2: blow-up of a point in a plane and sections of line bundles

We look at the fibers of Z^3 -> Z which (a,b,c) maps to (a+b-c). Each fiber has the structure of a module over the zero fiber (colored red). Alternatively, this is the result of grading a polynomial ring of three variables assigning them weights (1,1,-1). The planar strips are the monomials of the same grade: the red one are those of degree zero, the blue have negative grades, the green have positive grade. The red dots give generators of each strip as a module over the red one. The associated geometric picture is a blow up of a plane at a point: the polynomial ring is then the Cox ring of the blow up and the strips are sections of line bundles on the blow-up. In particular the red dots are sections of the respective line bundles over the exceptional set of the blow up.