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Dino Festi

Publications and source records attributed to Dino Festi.

15 recordsLinked to original sources
arXiv2025

Examples of real stable bundles on K3 surfaces

Motivated by gauge theory on manifolds with exceptional holonomy, we construct examples of stable bundles on K3 surfaces that are invariant under two involutions: one is holomorphic; and the other is anti-holomorphic. These bundles are obtained via the monad construction, and stability is examined using the Generalised Hoppe Criterion of Jardim-Menet-Prata-S\'a Earp, which requires verifying an arithmetic condition for elements in the Picard group of the surfaces. We establish this by using computer aid in two critical steps: first, we construct K3 surfaces with small Picard group-one branched double cover of $\mathbb{P}^1 \times \mathbb{P}^1$ with Picard rank $2$ using a new method which may be of independent interest; and second, we verify the arithmetic condition for carefully chosen elements of the Picard group, which provides a systematic approach for constructing further examples.

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arXiv2024

Polarized K3 surfaces with an automorphism of order 3 and low Picard number

In this paper, for each $d>0$, we study the minimum integer $h_{3,2d}\in \mathbb{N}$ for which there exists a complex polarized K3 surface $(X,H)$ of degree $H^2=2d$ and Picard number $ρ(X):=\textrm{rank } \textrm{Pic } X = h_{3,2d}$ admitting an automorphism of order $3$. We show that $h_{3,2}\in\{ 4,6\}$ and $h_{3,2d}=2$ for $d>1$. Analogously, we study the minimum integer $h^*_{3,2d}\in \mathbb{N}$ for which there exists a complex polarized K3 surface $(X,H)$ as above plus the extra condition that the automorphism acts as the identity on the Picard lattice of $X$. We show that $h^*_{3,2d}$ is equal to $2$ if $d>1$ and equal to $6$ if $d=1$. We provide explicit examples of K3 surfaces defined over $\mathbb{Q}$ realizing these bounds.

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arXiv2022

K3 surfaces with two involutions and low Picard number

Let $X$ be a complex algebraic K3 surface of degree $2d$ and with Picard number $\rho$. Assume that $X$ admits two commuting involutions: one holomorphic and one anti-holomorphic. In that case, $\rho \geq 1$ when $d=1$ and $\rho \geq 2$ when $d \geq 2$. For $d=1$, the first example defined over $\mathbb{Q}$ with $\rho=1$ was produced already in 2008 by Elsenhans and Jahnel. A K3 surface provided by Kond\={o}, also defined over $\mathbb{Q}$, can be used to realise the minimum $\rho=2$ for all $d\geq 2$. In these notes we construct new explicit examples of K3 surfaces over the rational numbers realising the minimum $\rho=2$ for $d=2,3,4$. We also show that a nodal quartic surface can be used to realise the minimum $\rho=2$ for infinitely many different values of $d$. Finally, we strengthen a result of Morrison by showing that for any even lattice $N$ of rank $1\leq r \leq 10$ and signature $(1,r-1)$ there exists a K3 surface $Y$ defined over $\mathbb{R}$ such that $\textrm{Pic} Y_\mathbb{C}=\textrm{Pic} Y \cong N$.

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arXiv2022

A Calabi-Yau threefold coming from two black holes

In this paper, we show that a set of six square roots of homogeneous polynomials in four variables, related to a binary system of black holes studied by Stefan Weinzierl, is not rationalizable. We prove it by showing that the variety $X$ associated to the product of four of the six square roots is not unirational. In particular, we show that the smooth model of $X$ is a Calabi-Yau threefold.

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arXiv2022

Rationalizability of field extensions with a view towards Feynman integrals

In 2021, Marco Besier and the first author introduced the concept of rationalizability of square roots to simplify arguments of Feynman integrals. In this work, we generalize the definition of rationalizability to field extensions. We then show that the rationalizability of a set of quadratic field extensions is equivalent to the rationalizability of the compositum of the field extensions, providing a new strategy to prove rationalizability of sets of square roots of polynomials.

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arXiv2021

Enriques involutions on pencils of K3 surfaces

The three pencils of K3 surfaces of minimal discriminant whose general element covers at least one Enriques surface are Kondō's pencils I and II, and the Apéry--Fermi pencil. We enumerate and investigate all Enriques surfaces covered by their general elements.

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arXiv2021

Counting elliptic fibrations on K3 surfaces

We solve the problem of counting jacobian elliptic fibrations on an arbitrary complex projective K3 surface up to automorphisms. We then illustrate our method with several explicit examples.

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arXiv2020

Rationalizability of square roots

Feynman integral computations in theoretical high energy particle physics frequently involve square roots in the kinematic variables. Physicists often want to solve Feynman integrals in terms of multiple polylogarithms. One way to obtain a solution in terms of these functions is to rationalize all occurring square roots by a suitable variable change. In this paper, we give a rigorous definition of rationalizability for square roots of ratios of polynomials. We show that the problem of deciding whether a single square root is rationalizable can be reformulated in geometrical terms. Using this approach, we give easy criteria to decide rationalizability in most cases of square roots in one and two variables. We also give partial results and strategies to prove or disprove rationalizability of sets of square roots. We apply the results to many examples from actual computations in high energy particle physics.

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arXiv2020

Arithmetic and geometry of a K3 surface emerging from virtual corrections to Drell--Yan scattering

We study a K3 surface, which appears in the two-loop mixed electroweak-quantum chromodynamic virtual corrections to Drell--Yan scattering. A detailed analysis of the geometric Picard lattice is presented, computing its rank and discriminant in two independent ways: first using explicit divisors on the surface and then using an explicit elliptic fibration. We also study in detail the elliptic fibrations of the surface and use them to provide an explicit Shioda--Inose structure. Moreover, we point out the physical relevance of our results.

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arXiv2018

Bhabha Scattering and a special pencil of K3 surfaces

We study a pencil of K3 surfaces that appeared in the $2$-loop diagrams in Bhabha scattering. By analysing in detail the Picard lattice of the general and special members of the pencil, we identify the pencil with the celebrated Apéry--Fermi pencil, that was related to Apéry's proof of the irrationality of $ζ(3)$ through the work of F. Beukers, C. Peters and J. Stienstra. The same pencil appears miraculously in different and seemingly unrelated physical contexts.

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arXiv2018

A practical algorithm to compute the geometric Picard lattice of K3 surfaces of degree $2$

Let $k$ be either a number a field or a function field over $\mathbb{Q}$ with finitely many variables. We present a practical algorithm to compute the geometric Picard lattice of a K3 surface over $k$ of degree $2$, i.e., a double cover of the projective plane over $k$ ramified above a smooth sextic curve. The algorithm might not terminate, but if it terminates then it returns a proven correct answer.

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arXiv2018

On the arithmetic of a family of degree-two K3 surfaces

Let $\mathbb{P}$ denote the weighted projective space with weights $(1,1,1,3)$ over the rationals, with coordinates $x,y,z,$ and $w$; let $\mathcal{X}$ be the generic element of the family of surfaces in $\mathbb{P}$ given by \begin{equation*} X\colon w^2=x^6+y^6+z^6+tx^2y^2z^2. \end{equation*} The surface $\mathcal{X}$ is a K3 surface over the function field $\mathbb{Q}(t)$. In this paper, we explicitly compute the geometric Picard lattice of $\mathcal{X}$, together with its Galois module structure, as well as derive more results on the arithmetic of $\mathcal{X}$ and other elements of the family $X$.

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arXiv2015

Unirationality of del Pezzo surfaces of degree two over finite fields

We prove that every del Pezzo surface of degree two over a finite field is unirational, building on the work of Manin and an extension by Salgado, Testa, and Várilly-Alvarado, who had proved this for all but three surfaces. Over general fields of characteristic not equal to two, we state sufficient conditions for a del Pezzo surface of degree two to be unirational.

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arXiv2012

The Cayley-Oguiso automorphism of positive entropy on a K3 surface

Recently Oguiso showed the existence of K3 surfaces that admit a fixed point free automorphism of positive entropy. The K3 surfaces used by Oguiso have a particular rank two Picard lattice. We show, using results of Beauville, that these surfaces are therefore determinantal quartic surfaces. Long ago, Cayley constructed an automorphism of such determinantal surfaces. We show that Cayley's automorphism coincides with Oguiso's free automorphism. We also exhibit an explicit example of a determinantal quartic whose Picard lattice has exactly rank two and for which we thus have an explicit description of the automorphism.

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