lefschetz-family 0.1.15

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). Here is a runtime benchmark for various examples, with an input precision of 1000 bits:

Hypersurface (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
Rational elliptic surface	10 seconds	300 digits
Elliptic K3 surface	30 seconds*	300 digits
Degree 2 K3 surface	5 minutes	300 digits

*for holomorphic periods

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.

Usage

Hypersurface

The first step is to define the polynomial $P$ defining the projective hypersurface $X=V(P)$. For instance, the following gives the Fermat elliptic curve:

R.<X,Y,Z> = PolynomialRing(QQ)
P = X**3+Y**3+Z**3

Then the following creates an object representing the hypersurface:

from lefschetz_family import Hypersurface
X = Hypersurface(P)

The period matrix of $X$ is the simply given by:

X.period_matrix

The module automatically uses available cores for computing numerical integrations and braids of roots. For this, the sage session needs to be made aware of the available cores. This can be done by adding the following line of code before launching the computation (replace 10 by the number of cores you want to use).

os.environ["SAGE_NUM_THREADS"] = '10'

See the computation of the periods of the Fermat quartic surface for a detailed usage example.

Copy-paste ready examples

The Fermat elliptic curve

os.environ["SAGE_NUM_THREADS"] = '10'
from lefschetz_family import Hypersurface
R.<X,Y,Z> = PolynomialRing(QQ)
P = X**3+Y**3+Z**3
X = Hypersurface(P, nbits=1500)
X.period_matrix

A quartic K3 surface of Picard rank 3

This one should take around 1 hour to compute, provided your computer has access to 10 cores.

os.environ["SAGE_NUM_THREADS"] = '10'
from lefschetz_family import Hypersurface
R.<W,X,Y,Z> = PolynomialRing(QQ)
P = (2*X*Y^2*Z + 3*X^2*Z^2 + 5*X*Y*Z^2 - 2*X*Z^3 + 2*Y*Z^3 + Z^4 + X^3*W - 3*X^2*Y*W - X*Y^2*W + Y^3*W - 2*X^2*Z*W - 2*Y^2*Z*W - 2*X*Z^2*W + 2*Y*Z^2*W - X^2*W^2 - X*Y*W^2 - 2*Y^2*W^2 - 2*X*Z*W^2 + 2*Y*W^3 - W^4)*2 + X^4 - Y^4 + Z^4 - W^4
fibration = [vector(ZZ, [10, -8, -2, 7]), vector(ZZ, [1, -1, 5, 10]), vector(ZZ, [-5, 7, 7, 10])]
X = Hypersurface(P, nbits=1200, fibration=fibration)

periods = X.holomorphic_period_matrix_modification

from lefschetz_family.numperiods.integerRelations import IntegerRelations
IR = IntegerRelations(X.holomorphic_period_matrix_modification)
# this is the rank of the transcendental lattice
transcendental_rank = X.holomorphic_period_matrix_modification.nrows()-IR.basis.rank()
# The Picard rank is thus
print("Picard rank:", 22-transcendental_rank)

Options

The object Hypersurface can be called with several options:

• nbits (positive integer, 400 by default): the number of bits of precision used as input for the computations. If a computation fails to recover the integral monodromy matrices, you should try to increase this precision. The output precision seems to be roughly linear with respect to the input precision.
• debug (boolean, False by default): whether coherence checks should be done earlier rather than late. We recommend setting to true only if the computation failed in normal mode.
• singular (boolean, False by default): whether the variety is singular. If it is (and in particular if the monodromy representation is not of Lefschetz type), the algorithm will try to desingularise the variety from the monodromy representation. This is work in progress.
• method ("voronoi" by default/"delaunay"/"delaunay_dual"): the method used for computing a basis of homotopy. voronoi uses integration along paths in the voronoi graph of the critical points; delaunay uses integration along paths along the delaunay triangulation of the critical points; delaunay_dual paths are along the segments connecting the barycenter of a triangle of the Delaunay triangulation to the middle of one of its edges. In practice, delaunay is more efficient for low dimension and low order varieties (such as degree 3 curves and surfaces, and degree 4 curves). This gain in performance is however hindered in higher dimensions because of the algebraic complexity of the critical points (which are defined as roots of high order polynomials, with very large integer coefficients). "delaunay" method is not working for now

Properties

The object Hypersurface has several properties. Fibration related properties, in positive dimension:

• fibration: a list of independant hyperplanes defining the iterative pencils. The first two element of the list generate the pencil used for the fibration.
• critical_values: the list critical values of that map.
• basepoint: the basepoint of the fibration (i.e. a non critical value).
• fibre: the fibre above the basepoint as a Hypersurface object.
• fundamental_group: the class computing representants of the fundamental group of $\mathbb P^1$ punctured at the critical values.
• paths: the list of simple loops around each point of critical_values. When this is called, the ordering of critical_values changes so that the composition of these loops is the loop around infinity.
• family: the one parameter family corresponding to the fibration.

Homology related properties:

• monodromy_matrices: the matrices of the monodromy action of paths on $H_{n-1}(X_b)$.
• vanishing_cycles: the vanshing cycles at each point of critical_values along paths.
• thimbles: the thimbles of $H_n(Y,Y_b)$. They are represented by a starting cycle in $H_n(Y_b)$ and a loop in $\mathbb C$ avoiding critical_values and pointed at basepoint.
• kernel_boundary: linear combinations of thimbles with empty boundary.
• extensions: integer linear combinations of thimbles with vanishing boundary.
• infinity_loops: extensions around the loop at infinity.
• homology_modification: a basis of $H_n(Y)$.
• intersection_product_modification: the intersection product of $H_n(Y)$.
• fibre_class: the class of the fibre in $H_n(Y)$.
• section: the class of a section in $H_n(Y)$.
• thimble_extensions: couples (t, T) such that T is the homology class in $H_n(Y)$ representing the extension of a thimble $\Delta \in H_{n-1}(X_b, X_{bb'})$ over all of $\mathbb P^1$, with $\delta\Delta =$t. Futhermore, the ts define a basis of the image of the boundary map $\delta$.
• invariant: the intersection of section with the fibre above the basepoint, as a cycle in $H_{n-2}({X_b}_{b'})$.
• exceptional_divisors: the exceptional cycles coming from the modification $Y\to X$, given in the basis homology_modification.
• homology: a basis of $H_n(X)$, given as its embedding in $H_2(Y)$.
• intersection_product: the intersection product of $H_n(X)$.
• lift: a map taking a linear combination of thimbles with zero boundary (i.e. an element of $\ker\left(\delta:H_n(Y, Y_b)\to H_{n-1}(Y_b)\right)$) and returning the homology class of its lift in $H_2(Y)$, in the basis homology_modification.
• lift_modification: a map taking an element of $H_n(Y)$ given by its coordinates in homology_modification, and returning its homology class in $H_n(X)$ in the basis homology.

Cohomology related properties:

• cohomology: a basis of $PH^n(X)$, represented by the numerators of the rational fractions.
• holomorphic_forms: the indices of the forms in cohomology that form a basis of holomorphic forms.
• picard_fuchs_equation(i): the picard fuchs equation of the parametrization of i-th element of cohomology by the fibration

Period related properties

• period_matrix: the period matrix of $X$ in the aforementioned bases homology and cohomology, as well as the cohomology class of the linear section in even dimension
• period_matrix_modification: the period matrix of the modification $Y$ in the aforementioned bases homology_modification and cohomology
• holomorphic_period_matrix: the periods of holomorphic_forms in the basis homology.
• holomorphic_period_matrix_modification: the periods of the pushforwards of holomorphic_forms in the basis homology_modification.

Miscellaneous properties:

• P: the defining equation of $X$.
• dim: the dimension of $X$.
• degree: the degree of $X$.
• ctx: the options of $X$, see related section above.

The computation of the exceptional divisors can be costly, and is not always necessary. For example, the Picard rank of a quartic surface can be recovered with holomorphic_period_matrix_modification alone.

EllipticSurface

Usage

The defining equation for the elliptic surface should be given as a univariate polynomial over a trivariate polynomial ring. The coefficients should be homogeneous of degree $3$.

R.<X,Y,Z> = PolynomialRing(QQ)
S.<t> = PolynomialRing(R)
P = X^2*Y+Y^2*Z+Z^2*X+t*X*Y*Z 

Then the following creates an object representing the surface:

from lefschetz_family import EllipticSurface
X = EllipticSurface(P)

Copy-paste ready examples

New rank records for elliptic curves having rational torsion, $\mathbb Z/2\mathbb Z$

We recover the result of Section 9 of New rank records for elliptic curves having rational torsion by Noam D. Elkies and Zev Klagsbrun.

os.environ["SAGE_NUM_THREADS"] = '10'

from lefschetz_family import EllipticSurface

R.<X,Y,Z> = QQ[]
S.<t> = R[]
U.<u> = S[]

A = (u^8 - 18*u^6 + 163*u^4 - 1152*u^2 + 4096)*t^4 + (3*u^7 - 35*u^5 - 120*u^3 + 1536*u)*t^3+ (u^8 - 13*u^6 + 32*u^4 - 152*u^2 + 1536)*t^2 + (u^7 + 3*u^5 - 156*u^3 + 672*u)*t+ (3*u^6 - 33*u^4 + 112*u^2 - 80)
B1 = (u^2 + u - 8)*t + (-u + 2)
B3 = (u^2 - u - 8)*t + (u^2 + u - 10)
B5 = (u^2 - 7*u + 8)*t + (-u^2 + u + 2)
B7 = (u^2 + 5*u + 8)*t + (u^2 + 3*u + 2)
B2 = -B1(t=-t,u=-u)
B4 = -B3(t=-t,u=-u)
B6 = -B5(t=-t,u=-u)
B8 = -B7(t=-t,u=-u)

P = -Y^2*Z + X^3 + 2*A*X^2*Z + product([B1, B2, B3, B4, B5, B6, B7, B8])*X*Z^2

surface = EllipticSurface(P(5), nbits=1000)
surface.mordell_weil

K3 surfaces and sphere packings

This example recovers the result of K3 surfaces and sphere packings by Tetsuji Shioda.

os.environ["SAGE_NUM_THREADS"] = '10'
from lefschetz_family import EllipticSurface

R.<X,Y,Z> = PolynomialRing(QQ)
S.<t> = PolynomialRing(R)

# you may modify these parameters
alpha = 3
beta = 5
n = 3

P = -Z*Y**2*t^n + X**3*t^n - 3*alpha*X*Z**2*t**n + (t**(2*n) + 1 - 2*beta*t**n)*Z^3

surface = EllipticSurface(P, nbits=1500)

# this is the Mordell-Weil lattice
surface.mordell_weil_lattice

# these are the types of the singular fibres
for t, _, n in surface.types:
    print(t+str(n) if t in ['I', 'I*'] else t)

Options

The options are the same as those for Hypersurface (see above).

Properties

The object EllipticSurface has several properties. Fibration related properties, in positive dimension:

• critical_values: the list critical values of that map.
• basepoint: the basepoint of the fibration (i.e. a non critical value).
• fibre: the fibre above the basepoint as a LefschetzFamily object.
• paths: the list of simple loops around each point of critical_points. When this is called, the ordering of critical_points changes so that the composition of these loops is the loop around infinity.
• family: the one parameter family corresponding to the fibration.

Homology related properties:

• extensions: the extensions of the fibration.
• extensions_morsification: the extensions of the morsification of the fibration.
• homology: the homology of $X$.
• singular_components: a list of lists of combinations of thimbles of the morsification, such that the elements of singular_components[i] form a basis of the singular components of the fibre above critical_values[i]. To get their coordinates in the basis homology, use X.lift(X.singular_components[i][j]).
• fibre_class: the class of the fibre in homology.
• section: the class of the zero section in homology.
• intersection_product: the intersection matrix of the surface in the basis homology.
• morsify: a map taking a combination of extensions and returning its coordinates on the basis of thimbles of the morsification.
• lift: a map taking a combination of thimbles of the morsification with empty boundary and returning its class in homology.
• types: types[i] is the type of the fibre above critical_values[i]. It is given as a triple t, M, nu where t is the letter of the type of the fibre ('I', 'II', etc.), M is the ${SL}_2(\mathbb Z)$ matrices so that M**(-1)*monodromy_matrices[i]*M is the representative $M_T$ of the monodromy class (see Table 1. of the paper), and nu is the multiplicity of the fibre in the cases where t is 'I' or 'I*' (otherwise nu is 1).

Cohomology related properties:

• holomorphic_forms: a basis of rational functions $f(t)$ such that $f(t) {Res}\frac{\Omega_2}{P_t}\wedge\mathrm dt$ is a holomorphic form of $S$.
• picard_fuchs_equations: the list of the Picard-Fuchs equations of the holomorphic forms mentionned previously.

Period related properties:

• period_matrix: the holomorphic periods of $X$ in the bases self.homology and self.holomorphic_forms.
• primary_periods: the holomorphic periods $X$ in the bases self.primary_lattice and self.holomorphic_forms

Sublattices of homology. Unless stated otherwise, lattices are given by the coordinates of a basis of the lattice in the basis homology:

• primary_lattice: The primary lattice of $X$, consisting of the concatenation of extensions, singular_components, fibre_class and section.
• neron_severi: the Néron-Severi group of $X$.
• trivial: the trivial lattice.
• essential_lattice: the essential lattice.
• mordell_weil: the Mordell-Weil group of $X$, described as the quotient module neron_severi/trivial.
• mordell_weil_lattice: the intersection matrix of the Mordell-Weil lattice of $X$.

Miscellaneous properties:

• ctx: the options of $X$, see related section above.

DoubleCover

Usage

The defining equation for the double cover should be given as a homogeneous polynomial of even degree. Such a polynomial $P$ represents the double cover $X = V(w^2-P)$.

R.<X,Y,Z> = PolynomialRing(QQ)
P = X^6+Y^6+Z^6

Then the following creates an object representing the variety:

from lefschetz_family import DoubleCover
X = DoubleCover(P)

Copy-paste ready examples

TODO

Options

The options are the same as those for Hypersurface (see above).

Properties

The object DoubleCover has several properties. Fibration related properties, in positive dimension:

• critical_values: the list critical values of that map.
• basepoint: the basepoint of the fibration (i.e. a non critical value).
• fibre: the fibre above the basepoint as a LefschetzFamily object.
• paths: the list of simple loops around each point of critical_points. When this is called, the ordering of critical_points changes so that the composition of these loops is the loop around infinity.
• family: the one parameter family corresponding to the fibration.

Homology related properties:

• extensions: the extensions of the fibration.
• homology: the homology of $X$.
• fibre_class: the class of the fibre in homology.
• section: the class of the zero section in homology.
• intersection_product: the intersection matrix of the surface in the basis homology.
• lift: a map taking a combination of thimbles of the morsification with empty boundary and returning its class in homology.

Cohomology related properties:

• holomorphic_forms: a basis of rational functions $f(t)$ such that $f(t) {Res}\frac{\Omega_2}{P_t}\wedge\mathrm dt$ is a holomorphic form of $S$.

Period related properties:

• period_matrix: the holomorphic periods of $X$ in the bases self.homology and self.holomorphic_forms.
• effective_periods: the holomorphic periods $X$ in the bases self.effective_lattice and self.holomorphic_forms

Sublattices of homology. Unless stated otherwise, lattices are given by the coordinates of a basis of the lattice in the basis homology:

• primary_lattice: The lattice of effective cycles of $X$, consisting of the concatenation of extensions, singular_components, fibre_class and section.

Miscellaneous properties:

• P: the defining equation of $X$.
• dim: the dimension of $X$.
• degree: the degree of $P$.
• ctx: the options of $X$, see related section above.

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.