Sporadic List of Terminal Fano 4-Fold Hypersurfaces in Weighted Projective Space of Index 1

This dataset contains a sporadic list of terminal Fano $4$-fold hypersurfaces of index $1$ in weighted projective space, discovered using fixed and dynamic (deep reinforcement learning) heuristic searches. 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 searches discover 121,980 distinct nonquasismooth terminal families, including 46,992 previously unknown families with $d>200$, reaching degrees as high as $3,234$. These extend beyond the range of the exhaustive classification for $d\leq200$, established in Terminal Fano 4-fold Hypersurfaces [4], with the complete classification archived separately on Zenodo [3]. Unlike an exhaustive classification, this list makes no claim of completeness at higher degrees. The database also incorporates the quasismooth terminal families classified by Gavin Brown and Alexander Kasprzyk [1], as recorded in the Graded Ring Database (GRDB) [2]. The search algorithms, including a neural network trained online using deep reinforcement learning, are developed in the accompanying paper Terminal Fano 4-fold Hypersurfaces via Deep Reinforcement Learning [6]. The methods for certifying terminality, including the analysis of hyperquotient singularities, are developed in the author's doctoral thesis [8] and the associated research. Data and format The list is provided as a SQL file, terminal_fano_4fold_index1_sporadic.sql, containing the table terminal_fano_4fold_index1_sporadic. 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 To illustrate the geometric information recorded for each family, 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 beyond the degree and weights of each family. Accessing the data The SQL file can be imported into SQLite using: sqlite3 terminal_fano_4fold_index1_sporadic.db ".read terminal_fano_4fold_index1_sporadic.sql" The resulting database can be queried using SQL or Python's sqlite3 module. For example, the following retrieves all nonquasismooth families with $d>200$: SELECT * FROM terminal_fano_4fold_index1_sporadic WHERE Equation_degree > 200 AND IsQsmooth = 0; The source code and experimental data associated with the heuristic searches are archived separately on Zenodo [7]. The exhaustive classification for $d\leq200$ and its corresponding source code are archived separately on Zenodo [3, 5]. 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, Classification of Terminal Fano 4-Fold Hypersurfaces in Weighted Projective Space of Index 1 up to Degree 200, Zenodo. https://doi.org/10.5281/zenodo.23836799. [4] Marc Truter, Terminal Fano 4-fold Hypersurfaces, in preparation. [5] Marc Truter, Terminal Fano 4-fold Hypersurfaces — Source Code, Version v1.0.0 [Computer software], Zenodo (2026). https://doi.org/10.5281/zenodo.23885752. [6] Marc Truter, Terminal Fano 4-fold Hypersurfaces via Deep Reinforcement Learning, arXiv:2603.15437. [7] Marc Truter, Terminal Fano 4-fold Hypersurfaces via Deep Reinforcement Learning — Source Code, Version v1.0.0 [Computer software], Zenodo (2026). https://doi.org/10.5281/zenodo.23886417. [8] 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].