lefschetz-family 0.1.19

This package implements algorithms relying on Lefschetz fibration theory to compute periods of algebraic varieties.

lefschetz-family

Description

This Sage package provides a means of efficiently computing periods of complex projective hypersurfaces and elliptic surfaces over $\mathbb P^1$ with certified rigorous precision bounds. It implements the methods described in

• Effective homology and periods of complex projective hypersurfaces (arxiv:2306.05263).
• A semi-numerical algorithm for the homology lattice and periods of complex elliptic surfaces over the projective line (arxiv:2401.05131).
• Periods of fibre products of elliptic surfaces and the Gamma conjecture (arxiv:2505.07685).
• Periods in algebraic geometry : computations and application to Feynman integrals (hal:tel-04823423).
• Galois groups of symmetric cubic surfaces (arxiv:2509.06785).

Please cite accordingly.

This package is a successor to the numperiods package by Pierre Lairez. It contains files taken from this package, that have sometimes been slightly modified to accomodate for new usage.

How to install

In a terminal, run

sage -pip install git+https://github.com/mkauers/ore_algebra.git
sage -pip install lefschetz-family

or

sage -pip install --user git+https://github.com/mkauers/ore_algebra.git
sage -pip install --user lefschetz-family

Alternatively, install the ore_alegbra package (available at https://github.com/mkauers/ore_algebra), then download this repository and add the path to the main folder to your sys.path.

Requirements

Sage 9.0 and above is recommended. Furthermore, this package has the following dependencies:

• Ore Algebra.
• The delaunay-triangulation package from PyPI.

Documentation

• Hypersurface for computing periods of hypersurfaces.
• EllipticSurface for computing periods of elliptic surfaces.
• DoubleCover for computing periods of ramified double cover of projective spaces.
• FibreProduct for computing periods of fibre products of elliptic surfaces.
• Fibration for computing monodromy representations of families of hypersurfaces.

Performance benchmarking

Here is a runtime benchmark for computing monodromy representations and periods of various types of varieties, with an input precision of 1000 bits:

Variety (generic)	Time (on 10 M1 cores)	Recovered precision (decimal digits)
Elliptic curve	5 seconds	340 digits
Quartic curve	90 seconds	340 digits
Quintic curve	5 minutes	340 digits
Sextic curve	30 minutes	300 digits
Cubic surface	40 seconds	340 digits
Quartic surface	1 hour	300 digits
Cubic threefold	15 minutes	300 digits
Cubic fourfold	20 hours	300 digits
Rational elliptic surface	10 seconds	N/A
Elliptic K3 surface	30 seconds*	300 digits
Degree 2 K3 surface	5 minutes	300 digits

*for holomorphic periods

Contact

For any question, bug or remark, please contact [e-mail].

Roadmap

Near future milestones:

• Encapsulate integration step in its own class
• Certified computation of the exceptional divisors
• Saving time on differential operator by precomputing cache before parallelization
• Computing periods of elliptic fibrations.
• Removing dependency on numperiods.

Middle term goals include:

• Making Delaunay triangulation functional again
• Having own implementation of 2D voronoi graphs/Delaunay triangulation

Long term goals include:

• Tackling cubic threefolds.
• Generic code for all dimensions.
• Computing periods of K3 surfaces with mildy singular quartic models.
• Dealing with other singularities, especially curves.
• Computing periods of complete intersections.
• Computing periods of weighted projective hypersurfaces, notably double covers of $\mathbb P^2$ ramified along a sextic.

Other directions include:

• Computation of homology through braid groups instead of monodromy of differential operators.

Project status

This project is actively being developped.