All Outputs (39)

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.

Seven new champion linear codes (2013)
Journal Article
Brown, G., & Kasprzyk, A. M. (2013). Seven new champion linear codes. LMS Journal of Computation and Mathematics, 16, https://doi.org/10.1112/S1461157013000041

We exhibit seven linear codes exceeding the current best known minimum distance d for their dimension k and block length n. Each code is defined over F₈, and their invariants [n,k,d] are given by [49,13,27], [49,14,26], [49,16,24], [49,17,23], [49,19... Read More about Seven new champion linear codes.

Small polygons and toric codes (2013)
Journal Article
Brown, G., & Kasprzyk, A. M. (2013). Small polygons and toric codes. Journal of Symbolic Computation, 51, https://doi.org/10.1016/j.jsc.2012.07.001

We describe two different approaches to making systematic classifications of plane lattice polygons, and recover the toric codes they generate, over small fields, where these match or exceed the best known minimum distance. This includes a [36,19,12]... Read More about Small polygons and toric codes.

Mirror symmetry and Fano manifolds (2013)
Presentation / Conference Contribution
Coates, T., Corti, A., Galkin, S., Golyshev, V., & Kasprzyk, A. M. (2013). Mirror symmetry and Fano manifolds.

We consider mirror symmetry for Fano manifolds, and describe how one can recover the classification of 3-dimensional Fano manifolds from the study of their mirrors. We sketch a program to classify 4-dimensional Fano manifolds using these ideas.

Fano polytopes (2012)
Book Chapter
Kasprzyk, A. M., & Nill, B. (2012). Fano polytopes. In A. Rebhan, L. Katzarkov, J. Knapp, R. Rashkov, & E. Scheidegger (Eds.), Strings, gauge fields, and the geometry behind: the legacy of Maximilian Kreuzer (349-364). World Scientific. https://doi.org/10.1142/9789814412551_0017

Fano polytopes are the convex-geometric objects corresponding to toric Fano varieties. We give a brief survey of classification results for different classes of Fano polytopes.

Reflexive polytopes of higher index and the number 12 (2012)
Journal Article
Kasprzyk, A. M., & Nill, B. (2012). Reflexive polytopes of higher index and the number 12. Electronic Journal of Combinatorics, 19(3), Article P9

We introduce reflexive polytopes of index l as a natural generalisation of the notion of a reflexive polytope of index 1. These l-reflexive polytopes also appear as dual pairs. In dimension two we show that they arise from reflexive polygons via a ch... Read More about Reflexive polytopes of higher index and the number 12.

Minkowski polynomials and mutations (2012)
Journal Article
Akhtar, M., Coates, T., Galkin, S., & Kasprzyk, A. M. (2012). Minkowski polynomials and mutations. Symmetry, Integrability and Geometry: Methods and Applications, 8, Article 094, pp. 707. https://doi.org/10.3842/SIGMA.2012.094

Given a Laurent polynomial f, one can form the period of f: this is a function of one complex variable that plays an important role in mirror symmetry for Fano manifolds. Mutations are a particular class of birational transformations acting on Lauren... Read More about Minkowski polynomials and mutations.

Roots of Ehrhart polynomials of smooth Fano polytopes (2011)
Journal Article
Hegedüs, G., & Kasprzyk, A. M. (2011). Roots of Ehrhart polynomials of smooth Fano polytopes. Discrete and Computational Geometry, 46(3), https://doi.org/10.1007/s00454-010-9275-y

V. Golyshev conjectured that for any smooth polytope P of dimension at most five, the roots $z\in\C$ of the Ehrhart polynomial for P have real part equal to -1/2. An elementary proof is given, and in each dimension the roots are described explicitly.... Read More about Roots of Ehrhart polynomials of smooth Fano polytopes.

The boundary volume of a lattice polytope (2011)
Journal Article
Hegedüs, G., & Kasprzyk, A. M. (2011). The boundary volume of a lattice polytope. Bulletin of the Australian Mathematical Society, 85(1), https://doi.org/10.1017/S0004972711002577

For a d-dimensional convex lattice polytope P, a formula for the boundary volume vol(δP) is derived in terms of the number of boundary lattice points on the first [d/2] dilations of P. As an application we give a necessary and sufficient condition fo... Read More about The boundary volume of a lattice polytope.

Canonical toric Fano threefolds (2010)
Journal Article
Kasprzyk, A. M. (2010). Canonical toric Fano threefolds. Canadian Journal of Mathematics, 62(6), https://doi.org/10.4153/CJM-2010-070-3

An inductive approach to classifying toric Fano varieties is given. As an application of this technique, we present a classification of the toric Fano threefolds with at worst canonical singularities. Up to isomorphism, there are 674,688 such varieti... Read More about Canonical toric Fano threefolds.

On the combinatorial classification of toric log del Pezzo surfaces (2010)
Journal Article
Kasprzyk, A. M., Kreuzer, M., & Nill, B. (2010). On the combinatorial classification of toric log del Pezzo surfaces. LMS Journal of Computation and Mathematics, 13, https://doi.org/10.1112/S1461157008000387

Toric log del Pezzo surfaces correspond to convex lattice polygons containing the origin in their interior and having only primitive vertices. An upper bound on the volume and on the number of boundary lattice points of these polygons is derived in t... Read More about On the combinatorial classification of toric log del Pezzo surfaces.

Bounds on fake weighted projective space (2009)
Journal Article
Kasprzyk, A. M. (2009). Bounds on fake weighted projective space. https://doi.org/10.2996/kmj/1245982903

A fake weighted projective space X is a Q-factorial toric variety with Picard number one. As with weighted projective space, X comes equipped with a set of weights (\lambda_0,...,\lambda_n). We see how the singularities of P(\lambda_0,...,\lambda_n)... Read More about Bounds on fake weighted projective space.

A Note on Palindromic δ -Vectors for Certain Rational Polytopes (2008)
Journal Article
Fiset, M., & Kasprzyk, A. M. (2008). A Note on Palindromic δ -Vectors for Certain Rational Polytopes. Electronic Journal of Combinatorics, 15, Article N18. https://doi.org/10.37236/893

Let P be a convex polytope containing the origin, whose dual is a lattice polytope. Hibi's Palindromic Theorem tells us that if P is also a lattice polytope then the Ehrhart δ-vector of P is palindromic. Perhaps less well-known is that a similar resu... Read More about A Note on Palindromic δ -Vectors for Certain Rational Polytopes.

Toric Fano three-folds with terminal singularities (2006)
Journal Article
Kasprzyk, A. M. (2006). Toric Fano three-folds with terminal singularities. TÔhoku Mathematical Journal, 58(1), 101-121. https://doi.org/10.2748/tmj/1145390208

This paper classifies all toric Fano 3-folds with terminal singularities. This is achieved by solving the equivalent combinatoric problem; that of finding, up to the action of GL(3,Z), all convex polytopes in Z^3 which contain the origin as the only... Read More about Toric Fano three-folds with terminal singularities.

Interactions with Lattice Polytopes
Presentation / Conference Contribution
(2017, September). Interactions with Lattice Polytopes. Presented at Interactions with Lattice Polytopes, Magdeburg, Germany

This book collects together original research and survey articles highlighting the fertile interdisciplinary applications of convex lattice polytopes in modern mathematics. Covering a diverse range of topics, including algebraic geometry, mirror symm... Read More about Interactions with Lattice Polytopes.

On the Fine Interior of Three-Dimensional Canonical Fano Polytopes
Presentation / Conference Contribution
Batyrev, V., Kasprzyk, A., & Schaller, K. (2017, September). On the Fine Interior of Three-Dimensional Canonical Fano Polytopes. Presented at Interactions with Lattice Polytopes, Magdeburg, Germany

The Fine interior ∆FI of a d-dimensional lattice polytope ∆ is a rational subpolytope of ∆ which is important for constructing minimal birational models of non-degenerate hypersurfaces defined by Laurent polynomials with Newton polytope ∆. This paper... Read More about On the Fine Interior of Three-Dimensional Canonical Fano Polytopes.

Toric Sarkisov Links of Toric Fano Varieties
Presentation / Conference Contribution
Brown, G., Buczyński, J., & Kasprzyk, A. (2019, April). Toric Sarkisov Links of Toric Fano Varieties. Presented at Birational Geometry, Kähler–Einstein Metrics and Degenerations, Moscow, Shanghai and Pohang

We explain a web of Sarkisov links that overlies the classification of Fano weighted projective spaces in dimensions 3 and 4, extending results of Prokhorov.

Machine learning detects terminal singularities
Presentation / Conference Contribution
Kasprzyk, A. M., Coates, T., & Veneziale, S. (2023, December). Machine learning detects terminal singularities. Presented at 37th Conference on Neural Information Processing Systems (NeurIPS 2023), New Orleans, USA

Algebraic varieties are the geometric shapes defined by systems of polynomial equations; they are ubiquitous across mathematics and science. Amongst these algebraic varieties are Q-Fano varieties: positively curved shapes which have Q-factorial termi... Read More about Machine learning detects terminal singularities.