A python package for studying complete intersections in products of projective spaces. There is also some basic functionality for studying branched covers. The goal is to carry out some quick computations and search for interesting examples. There are many existing packages which deal with varieties in a single projective space, and looking at varieties in products yields more examples.
Currently, the package can compute Chern numbers of these varieties. It can also compute Chern numbers of branched covers when the branch locus is another complete intersection.
Users are warned that the computational results have not been rigorously tested, only checked against examples which I've worked out by hand. If you spot any errors, or would be interested in adding some functionality to the code base please contact me at my gmail address.
To use the package, launch python from the root of this directory.
You'll need to have the sympy python package installed or use a virtual environment.
See the installation details below.
For documentation of the functions, you'll have to look at the code and follow these examples for now :)
$ python
>>> # Import the package
>>> import mpci
>>> # First make a P1 x P2 and a P2 and look at their total Chern classes.
>>> p21 = mpci.MultiProj([1,2])
>>> p12.chern
Poly(6*x_0*x_1**2 + 6*x_0*x_1 + 2*x_0 + 3*x_1**2 + 3*x_1 + 1, x_0, x_1, domain='ZZ')
>>> p2 = mpci.MultiProj([2])
>>> p2.chern
Poly(3*x_0**2 + 3*x_0 + 1, x_0, domain='ZZ')
>>> # Then get all of the Chern numbers of a divisor of degree (1,2)
>>> # This is a degree=5 del Pezzo, which is P2 blown up at 4 points.
>>> mpci.CompIntersection(p21, [[1,2]]).get_all_chern_numbers()
{(1, 1): 5, (2,): 7}
>>> # Here's a del Pezzo double plane we find by double branching P2 over a deg=4 hypersurface
>>> mpci.CompIntersection(p2, []).get_chern_numbers_branched([4], 2)
{(1, 1): 2, (2,): 10}
>>> # Compute chern numbers of a divisor of degree (2,1) inside of P1 x P2
>>> mpci.CompIntersection(p12, [[2, 1]]).get_all_chern_numbers()
{(1, 1): 8, (2,): 4}
>>> # Compute chern numbers of P2 branched over a degree 2 curve. We'll get the same result as above.
>>> mpci.CompIntersection(p2, []).get_chern_numbers_branched([2], 2)
{(1, 1): 8, (2,): 4}
>>> # Here we make a product of three P1s and a P2 x P2
>>> p111 = mpci.MultiProj([1,1,1])
>>> p22 = mpci.MultiProj([2,2])
>>> # We find degree=6 del Pezzo surfaces in each
>>> mpci.CompIntersection(p111, [[1, 1, 1]]).get_all_chern_numbers()
{(1, 1): 6, (2,): 6}
>>> mpci.CompIntersection(p22, [[1, 1], [1,1]]).get_all_chern_numbers()
{(1, 1): 6, (2,): 6}
>>> # View the second as a subvariety of the deg=(1,1) hypersurface in P2 x P2.
>>> # Now let's increase the (multi)degrees of these complete intersections...
>>> # First multiply by 2, yielding K3 surfaces
>>> mpci.CompIntersection(p111, [[2,2,2]]).get_all_chern_numbers()
{(1, 1): 0, (2,): 24}
>>> mpci.CompIntersection(p22, [[1, 1], [2,2]]).get_all_chern_numbers()
{(1, 1): 0, (2,): 24}
>>> # Now we'll multiply by 3
>>> mpci.CompIntersection(p111, [[3,3,3]]).get_all_chern_numbers()
{(1, 1): 18, (2,): 90}
>>> mpci.CompIntersection(p22, [[1, 1], [3,3]]).get_all_chern_numbers()
{(1, 1): 18, (2,): 90}
The project only requires sympy. To install it in a virtual environment...
$ python3 -m pipenv install Pipfile
You only need to run the above upon first installation and might want to change the python version in Pipfile to ensure that this can run on your computer. Then to run the virtual environment...
$ python3 -m pipenv shell