Classification of Terminal Fano 4-Fold Hypersurfaces in Weighted Projective Space of Index 1 up to Degree 200

This dataset contains the classification, up to weighted degree $d\leq200$, of terminal Fano $4$-fold hypersurfaces of index $1$ in weighted projective space. We consider general hypersurfaces $$X_d\subset\mathbb{P}(a_1,\ldots,a_6),\qquad d=\sum_{i=1}^{6}a_i-1,$$ with at worst terminal singularities. The classification includes 77,353 nonquasismooth families, extending beyond the quasismooth classification of Gavin Brown and Alexander Kasprzyk [1]. The database also incorporates their quasismooth terminal families within the same degree bound, as recorded in the Graded Ring Database (GRDB) [2]. The classification and methods for certifying terminality, including the analysis of hyperquotient singularities, are developed in the author's doctoral thesis [5] and the accompanying paper Terminal Fano 4-fold Hypersurfaces [3]. Data and format The classification is provided as a SQL file, terminal_fano_4fold_index1_200.sql, containing the table terminal_fano_4fold_index1_200. Each record represents a family of hypersurfaces and includes: Equation_degree: The weighted degree $d$ of the defining equation. a1, ..., a6: The six weights of the ambient weighted projective space. CanonicalClass: The integer $d-\sum a_i=-1$. Degree: The exact anticanonical volume $(-K_X)^4=d/(a_1\cdots a_6)$, expressed as a rational number. Degree_dec: The decimal approximation of the anticanonical volume. Type_I_Point, Type_II_Point, Type_III_Point: The singularity baskets of points, organised by local singularity type. Type_I_Curve, Type_II_Curve, Type_III_Curve: The corresponding singularity baskets of curves. Point_Basket_size, Curve_Basket_size: The numbers of point and curve singularity entries, counted with multiplicity. IsSmooth, IsQsmooth, IsTerminal, IsCanonical: Indicators of the corresponding geometric properties. Example Consider the nonquasismooth terminal Fano hypersurface $$X_{136}\subset\mathbb{P}(1,1,5,25,37,68).$$ The database records two singular point entries, $$\frac{1}{37}(1,1,5,31),\qquad\frac{1}{25}(1,1,5,12,18;11),$$ and a singular curve entry $$\frac{1}{5}(0,1,2,3).$$ Its anticanonical volume is $$(-K_X)^4=\frac{136}{1\cdot1\cdot5\cdot25\cdot37\cdot68}=\frac{2}{4625}.$$ This illustrates the geometric information available for each family beyond its degree and weights. Accessing the data The SQL file can be imported into SQLite using: sqlite3 terminal_fano_4fold_index1_200.db ".read terminal_fano_4fold_index1_200.sql" The resulting database can be queried using SQL or Python's sqlite3 module. For example, the following retrieves all nonquasismooth families of degree $200$: SELECT * FROM terminal_fano_4fold_index1_200 WHERE Equation_degree = 200 AND IsQsmooth = 0; The source code used to generate and analyse the classification is archived separately on Zenodo [4]. Acknowledgements This work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) under grant number EP/W523793/1. References [1] Gavin Brown and Alexander M. Kasprzyk, Four-dimensional projective orbifold hypersurfaces, Experimental Mathematics 25 (2016), no. 2, 176–193. https://doi.org/10.1080/10586458.2015.1054054. [2] Gavin Brown and Alexander Kasprzyk, Graded Ring Database. [3] Marc Truter, Terminal Fano 4-fold Hypersurfaces, in preparation. [4] Marc Truter, Terminal Fano 4-fold Hypersurfaces — Source Code, Version v1.0.0 [Computer software], Zenodo (2026). https://doi.org/10.5281/zenodo.23885752. [5] Marc Truter, Terminal Hypersurfaces in Weighted Projective Spaces, PhD thesis, University of Warwick. If using this dataset, please cite the Zenodo record and the associated mathematical work. If using the quasismooth classification data, please also cite Brown and Kasprzyk [1].