How to use JavaView

You need the following

I expect that you know it, but just in case let me give this intruction: If you use full version of JavaView click right bottom 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, so that you can verify the argument.

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 pitures are obtained by merging simple pictures, with possible adjustements 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 0: blow-up of a point in a plane as a Proj of graded ring

The picture presents 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.

Example 1: two convex functions defining two resolutions of a singularity

This example explains two resolutions of the 3-dimensional singularity z_1z_2z_3=w^2 in terms of two convex functions on a triangle which is a section of a 3-dim cone. The function whose graph is on the left determines a resolution which has three flopping curve (value at f_i's = 1 and at e_i'1= 2) and on the right is what we get once we flop one of them (change the value at f_1 to 2.5):

Example 1A: resolving products, getting 4dim symplectic resolutions

Here are product resolutions of A_1xA_1 and A_2xA_2. Note Z_2 symmetries which interchange the factors in the resolution.

Example 2: two convex functions defining two SQM models of a Fano space

The blow-up of P^3 at two points has one flopping curve: the strict transform of the line passing through them. The green shade is the convex hull of the generators of rays. Inside, the blue polytope indicates the values of the function defining the particular fan. The first picture determines the fan related to the blow-up of P^3, value at e_i is 1 at f_1 and f_2 is 1/2. After we flop we take value at all points 1 except of e_0 and e_3 where we set 1/2. Rotate the picture so that you see two projections to first Hirzebruch surface and a projection to P^1.

Example 3: SQM models of Fano space, P^3 blown up in 4 points

First we look at the fan of P^3 blown up at four points

Next take three flopping curves which are strict trasforms of lines passing two blow-up points and flop them together. 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).

Example 4: P^3 blown up in 3 points: decomposition of movable cone into Mori chambers associated to SQM models

Here is the decomposition of movable cone of P^3 blown up in 3 points. The biggest white chamber is assciated to nef cone of the blow up. The purple has a contraction to P^1xP^1xP^1. There are also 3 chambers of green and 3 chambers of light-blue type (we put only one of each type to make the picture transparent enough. Observe that the division is made by hyperplane division.