Database

In this repository, we collect data on line bundle cohomologies over curves in a del Pezzo surface of degree 6. This data was analyzed in our publication using Decision Trees. The so-obtained insights, supplemented and extended by a working understanding of the corresponding algebraic geometry, lead to the software H0Approximator.

How to read the CSV files

We have computed line bundle cohomologies for line bundles of different degrees over curves of different genera. The curves and line bundles are specified as follows: A del Pezzo surface is a complex two-dimensional surface obtained from blowing up the complex projective space CP^2 in three points in general position. We denote the hyperplane class of CP^2 by H and the divisor classes of the three blowup divisors by E_1, E_2, and E_3, respectively. A curve inside the del Pezzo surface given as a section of a H + b E_1 + c E_2 + d E_3 is then specified as D_C=(a;b,c,d). Similarly, we specify a line bundle L by giving the first Chern class of the bundle for each divisor class, D_L=(a';b',c',d'). The folders in this repository are organized by the genus of the curve, with subfolders for the individual line bundles.

Folder structure

Folder	Description
Genus-1	A genus one curve with D_C=(3;-1,-1,-1). This curve has 7 monomials. We compute data for 13 different line bundles.
Genus-1-2	A disjoint union of a genus 0 and genus 2 curve genus one curve with D_C=(4;-1,-2,1). This curve has 11 monomials. We compute data for 13 different line bundles.
Genus-2	A genus two curve with D_C=(4;-1,-2,-1). This curve has 10 monomials. We compute data for 13 different line bundles.
Genus-2-2	A genus two curve with D_C=(4;-1,-2,0). This curve has 11 monomials. We compute data for 1 line bundle.
Genus-2-3	Two genus two curves. The first is D_C=(3;-3,-1,-2). This curve has 10 monomials. We compute data for 1 line bundle. The second is D_C=(4;-7,-1,-3). This curve has 10 monomials. We compute data for 1 line bundle.
Genus-3	A genus three curve with D_C=(4;-1,-1,-1). This curve has 12 monomials. We compute data for 13 different line bundles.
Genus-4	A genus four curve with D_C=(5;-2,-2,-1). This curve has 14 monomials. We compute data for 13 different line bundles.
Genus-5	A genus five curve with D_C=(5;-1,-1,-2). This curve has 16 monomials. We compute data for 13 different line bundles.
Genus-6	A genus five curve with D_C=(6;-3,-2,-1). This curve has 18 monomials. We compute data for 2 different line bundless.

Each folder contains a file CurveData.csv and subfolders (one for each line bundle). Each subfolder will contain a SummaryOfResults folder with a file ResultsWithSplitIntersectionsPlusLocalSections.csv.

Structure of CurveData.csv

Here we explain what the different columns of the CurveData.csv file mean:

Column	Description
coeffs	Array specifying the values of the coefficients that appear in front of the monomials that define the curve inside the del Pezzo surface. We compute data for all c_i in {0,1} (except for the Genus-2-3 curves, where we take them from {-1,0,1}), a curve with m monomials has 2^(m) different combinations of coefficients. We exclude the case where all coefficients are zero
curve-smooth	TRUE if the curve with the corresponding coefficients is smooth, FALSE otherwise
curve-split-number	The number of components the curve splits into
splits-reduced	For each component that the curves splits into, this specifies whether the component is reduced (True) or not (False)
splits-smooth	For each component that the curves splits into, this specifies whether the component is smooth (True) or not (False)
splits-genera	For each component that the curves splits into, this specifies genus of the split component
splits-intersections	This is the intersection matrix of the split components
splits-classes	This assigns classes to the different split types. This data is used to check whether different splits actually agree
split-type	An integer labeling the type of splits built from various topological invariants. See the paper for more details
determinant	The determinant of the intersection matrix. This is, up to a sign, permutation invariant and is used to distinguish different splits
split-type2	Another integer labeling the type of splits built from various topological invariants. We used this integer for the decision trees. See the paper for more details

Structure of ResultsWithSplitIntersectionsPlusLocalSections.csv

Here we explain what the different columns of the ResultsWithSplitIntersectionsPlusLocalSections.csv file mean:

Column	Description
coeffs	Array specifying the values of the coefficients that appear in front of the monomials that define the curve inside the del Pezzo surface. We compute data for all c_i in {0,1}
curve	Specification of the curve D_C=(a;b,c,d) as explained above
bundle	Specification of the bundle D_L=(a';b',c',d') as explained above
time	Computation time of the bundle cohomology
splits	The number of components the curve splits into
h0	The dimension of the zeroth cohomology group of the bundle on the curve
h1	The dimension of the first cohomology group of the bundle on the curve
SplitIntersections	Intersections of the line bundle with the split components of the curve
rho	Integer lower bound on the dimension of h0 computed using Brill-Noether theory
generic	Indicates whether the cohomology is generic, i.e. whether h0 and h1 have their minimal values (indicated by a 1) or not (indicated by a 0)
local-sections-A	The values of [h0,h1] for each split component of the curve. 'NA' means we have not computed the value individually

Citation

If you want to cite this data, please use the bibtex below or download it here:

@article{Bies:2020xxx,
    author = "Bies, Martin and Cveti{\v c}, Mirjam and Donagi, Ron and Lin, Ling and Liu, Muyang and Ruehle, Fabian",
    title = "{Machine Learning and Algebraic Approaches towards Complete Matter Spectra in 4d F-theory}",
    eprint = "2007.00009",
    archivePrefix = "arXiv",
    primaryClass = "hep-th",
    reportNumber = "UPR-1305-T, CERN-TH-2020-111",
    month = "6",
    year = "2020"
}

@Misc{Database,
  author       = "Bies, Martin and Cveti{\v c}, Mirjam and Donagi, Ron and Lin, Ling and Liu, Muyang and Ruehle, Fabian",
  howpublished = {\url{https://github.com/Learning-line-bundle-cohomology/Database}},
  title        = {Database},
  year         = "2020"
}