All Outputs (31)

On K-moduli of quartic threefolds (2024)
Journal Article

The family of smooth Fano 3-folds with Picard rank 1 and anticanonical volume 4 consists of quartic 3-folds and of double covers of the 3-dimensional quadric branched along an octic surface. They can all be parametrised as complete intersections of a... Read More about On K-moduli of quartic threefolds.

Polytopes and machine learning (2023)
Journal Article

We introduce machine learning methodology to the study of lattice polytopes. With supervised learning techniques, we predict standard properties such as volume, dual volume, reflexivity, etc, with accuracies up to 100%. We focus on 2d polygons and 3d... Read More about Polytopes and machine learning.

Machine learning the dimension of a Fano variety (2023)
Journal Article

Fano varieties are basic building blocks in geometry – they are ‘atomic pieces’ of mathematical shapes. Recent progress in the classification of Fano varieties involves analysing an invariant called the quantum period. This is a sequence of integers... Read More about Machine learning the dimension of a Fano variety.

On the maximum dual volume of a canonical Fano polytope (2022)
Journal Article

We give an upper bound on the volume vol(P*) of a polytope P* dual to a d-dimensional lattice polytope P with exactly one interior lattice point, in each dimension d. This bound, expressed in terms of the Sylvester sequence, is sharp, and is achieved... Read More about On the maximum dual volume of a canonical Fano polytope.

Databases of quantum periods for Fano manifolds (2022)
Journal Article

Fano manifolds are basic building blocks in geometry - they are, in a precise sense, atomic pieces of shapes. The classification of Fano manifolds is therefore an important problem in geometry, which has been open since the 1930s. One can think of th... Read More about Databases of quantum periods for Fano manifolds.

Laurent polynomials in Mirror Symmetry: why and how? (2022)
Journal Article

We survey the approach to mirror symmetry via Laurent polynomials, outlining some of the main conjectures, problems, and questions related to the subject. We discuss: how to construct Landau-Ginzburg models for Fano varieties; how to apply them to cl... Read More about Laurent polynomials in Mirror Symmetry: why and how?.

Hilbert series, machine learning, and applications to physics (2022)
Journal Article

We describe how simple machine learning methods successfully predict geometric properties from Hilbert series (HS). Regressors predict embedding weights in projective space to ∼1 mean absolute error, whilst classifiers predict dimension and Gorenstei... Read More about Hilbert series, machine learning, and applications to physics.

Maximally mutable Laurent polynomials (2021)
Journal Article

We introduce a class of Laurent polynomials, called maximally mutable Laurent polynomials (MMLPs), which we believe correspond under mirror symmetry to Fano varieties. A subclass of these, called rigid, are expected to correspond to Fano varieties wi... Read More about Maximally mutable Laurent polynomials.

Laurent inversion (2019)
Journal Article

We describe a practical and effective method for reconstructing the deformation class of a Fano manifold X from a Laurent polynomial f that corresponds to X under Mirror Symmetry. We explore connections to nef partitions, the smoothing of singular to... Read More about Laurent inversion.

Gorenstein Formats, Canonical and Calabi–Yau Threefolds (2019)
Journal Article

Gorenstein formats present the equations of regular canonical, Calabi–Yau and Fano varieties embedded by subcanonical divisors. We present a new algorithm for the enumeration of these formats based on orbifold Riemann-Roch and knapsack packing-type a... Read More about Gorenstein Formats, Canonical and Calabi–Yau Threefolds.

Ehrhart polynomial roots of reflexive polytopes (2019)
Journal Article

Recent work has focused on the roots z∈C of the Ehrhart polynomial of a lattice polytope P. The case when Rz=−1/2 is of particular interest: these polytopes satisfy Golyshev's "canonical line hypothesis". We characterise such polytopes when dim(P)≤7.... Read More about Ehrhart polynomial roots of reflexive polytopes.

Quantum Periods For Certain Four-Dimensional Fano Manifolds (2018)
Journal Article
Coates, T., Galkin, S., Kasprzyk, A., & Strangeway, A. (2018). Quantum Periods For Certain Four-Dimensional Fano Manifolds. Experimental Mathematics, 29(2), 183-221. https://doi.org/10.1080/10586458.2018.1448018

We collect a list of known four-dimensional Fano manifolds and compute their quantum periods. This list includes all four-dimensional Fano manifolds of index greater than one, all four-dimensional toric Fano manifolds, all four-dimensional products o... Read More about Quantum Periods For Certain Four-Dimensional Fano Manifolds.

Fano 3-folds in P2xP2 format, Tom and Jerry (2017)
Journal Article
Brown, G., Kasprzyk, A. M., & Qureshi, I. (2018). Fano 3-folds in P2xP2 format, Tom and Jerry. European Journal of Mathematics, 4(1), 51-72. https://doi.org/10.1007/s40879-017-0200-2

We study Q-factorial terminal Fano 3-folds whose equations are modelled on those of the Segre embedding of P^2xP^2. These lie in codimension 4 in their total anticanonical embedding and have Picard rank 2. They fit into the current state of classific... Read More about Fano 3-folds in P2xP2 format, Tom and Jerry.

Minimality and mutation-equivalence of polygons (2017)
Journal Article
Kasprzyk, A. M., Nill, B., & Prince, T. (in press). Minimality and mutation-equivalence of polygons. Forum of Mathematics, Sigma, 5(e18), https://doi.org/10.1017/fms.2017.10

We introduce a concept of minimality for Fano polygons. We show that, up to mutation, there are only finitely many Fano polygons with given singularity content, and give an algorithm to determine representatives for all mutation-equivalence classes o... Read More about Minimality and mutation-equivalence of polygons.

Quantum periods for 3-dimensional Fano manifolds (2016)
Journal Article
Coates, T., Corti, A., Galkin, S., & Kasprzyk, A. M. (2016). Quantum periods for 3-dimensional Fano manifolds. Geometry and Topology, 20(1), https://doi.org/10.2140/gt.2016.20.103

The quantum period of a variety X is a generating function for certain Gromov-Witten invariants of X which plays an important role in mirror symmetry. In this paper we compute the quantum periods of all 3-dimensional Fano manifolds. In particular we... Read More about Quantum periods for 3-dimensional Fano manifolds.

Mirror symmetry and the classification of orbifold del Pezzo surfaces (2016)
Journal Article

We state a number of conjectures that together allow one to classify a broad class of del Pezzo surfaces with cyclic quotient singularities using mirror symmetry. We prove our conjectures in the simplest cases. The conjectures relate mutation-equival... Read More about Mirror symmetry and the classification of orbifold del Pezzo surfaces.

Four-dimensional projective orbifold hypersurfaces (2015)
Journal Article
Brown, G., & Kasprzyk, A. M. (2015). Four-dimensional projective orbifold hypersurfaces. Experimental Mathematics, 25(2), https://doi.org/10.1080/10586458.2015.1054054

We classify four-dimensional quasismooth weighted hypersurfaces with small canonical class, and verify a conjecture of Johnson and Kollar on infinite series of quasismooth hypersurfaces with anticanonical hyperplane section in the case of fourfolds.... Read More about Four-dimensional projective orbifold hypersurfaces.

Mutations of Fake Weighted Projective Planes (2015)
Journal Article
Akhtar, M. E., & Kasprzyk, A. M. (2016). Mutations of Fake Weighted Projective Planes. Proceedings of the Edinburgh Mathematical Society, 59(2), 271-285. https://doi.org/10.1017/S0013091515000115

In previous work by Coates, Galkin, and the authors, the notion of mutation between lattice polytopes was introduced. Such a mutation gives rise to a deformation between the corresponding toric varieties. In this paper we study one-step mutations tha... Read More about Mutations of Fake Weighted Projective Planes.

Four-dimensional Fano toric complete intersections (2015)
Journal Article
Coates, T., Kasprzyk, A. M., & Prince, T. (2015). Four-dimensional Fano toric complete intersections. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 471, https://doi.org/10.1098/rspa.2014.0704

We find at least 527 new four-dimensional Fano manifolds, each of which is a complete intersection in a smooth toric Fano manifold.

Mutations of fake weighted projective spaces (2014)
Journal Article
Coates, T., Gonshaw, S., Kasprzyk, A. M., & Nabijou, N. (2014). Mutations of fake weighted projective spaces. Electronic Journal of Combinatorics, 21(4), Article P4.14

We characterise mutations between fake weighted projective spaces, and give explicit formulas for how the weights and multiplicity change under mutation. In particular, we prove that multiplicity-preserving mutations between fake weighted projective... Read More about Mutations of fake weighted projective spaces.