arXiv subjects

Giulia Gugiatti

Publications and source records attributed to Giulia Gugiatti.

7 recordsLinked to original sources

arXiv2026

Geometric realisation of hypergeometric local systems

We prove a realisation theorem for irreducible hypergeometric local systems defined over the rational numbers in terms of families of affine varieties in algebraic tori. The families we consider have been studied extensively in the literature and appear in mirror symmetry. Our result holds unconditionally for families with one-dimensional or even-dimensional fibres. It holds under a monodromy assumption for families with fibres of odd dimension greater than one.

math.AG↗

arXiv2025

On the mirrors of low-degree del Pezzo surfaces

We compare different constructions of mirrors of del Pezzo surfaces, focusing on degree $d \leq 3$. In particular, we extract Lefschetz fibrations, with associated exceptional collections, from the mirrors obtained via the Hori-Vafa and Fanosearch program constructions, which we relate to one another. We show with geometric methods that the Lefschetz fibrations define categorical mirrors. With a more explicit approach, we give a sequence of (numerical) mutations relating the exceptional collections considered by Auroux, Katzarkov, and Orlov with those arising in this paper. This uses the theory of surface-like pseudolattices, and extends some of the string junction results of Grassi, Halverson and Shaneson. Our argument lifts directly to an equivalence of certain Fukaya-Seidel categories arising from our fibrations and those of Auroux, Katzarkov, and Orlov.

math.AG↗

arXiv2024

Hypergeometric local systems over $\mathbb{Q}$ with Hodge vector $(1,1,1,1)$

We consider all irreducible rank-4 hypergeometric local systems defined over $\mathbb{Q}$ that support a rational one-dimensional variation of Hodge structures of weight 3 and Hodge vector $(1,1,1,1)$. Up to a natural equivalence there are only 47 cases. The first 14 cases have maximally unipotent monodromy at one point and have been extensively studied in the literature. We show that all 47 local systems are associated to families of generically smooth threefolds and we analyze the geometry and arithmetic at their conifold point.

math.AG↗

arXiv2023

Full exceptional collections for anticanonical log del Pezzo surfaces

Motivated by homological mirror symmetry, this paper constructs explicit full exceptional collections for the canonical stacks associated with the series of log del Pezzo surfaces constructed by Johnson and Kollár. These surfaces have cyclic quotient, non-Gorenstein, singularities. The construction involves both the $\mathrm{GL}(2,\mathbb{C})$ McKay correspondence, and the study of the minimal resolutions of the surfaces, which are birational to degree two del Pezzo surfaces. We show that a degree two del Pezzo surface arises in this way if and only if it admits a generalized Eckardt point, and in the course of the paper we classify the blow-ups of $\mathbb{P}^2$ giving rise to them. Our result on the adjoints of the functor of Ishii-Ueda applies to any finite small subgroup of $\mathrm{GL}(2,\mathbb{C})$.

math.AG↗

arXiv2023

Reflexive polygons and rational elliptic surfaces

In this note we study in detail the geometry of eight rational elliptic surfaces naturally associated to the sixteen reflexive polygons. The elliptic fibrations supported by these surfaces correspond under mirror symmetry to the eight families of smooth del Pezzo surfaces with very ample anticanonical bundle.

math.AG↗

arXiv2022

A Prym Hypergeometric

We study a hypergeometric local system that arises from the quantum Chen-Ruan cohomology of a family of weighted del Pezzo hypersurfaces. We prove that it is the anti-invariant variation of a pencil of genus-7 curves with respect to an involution having 4 fixed points.

math.AG↗

arXiv2019

Hyperelliptic Integrals and Mirrors of the Johnson-Kollár del Pezzo Surfaces

For all k>0 integer, we consider the regularised I-function of the family of del Pezzo surfaces of degree 8k+4 in P(2,2k+1,2k+1, 4k+1), first constructed by Johnson and Kollár. We show that this function, which is of hypergeometric type, is a period of an explicit pencil of curves. Thus the pencil is a candidate LG mirror of the family of del Pezzo surfaces. The main feature of these surfaces, which makes the mirror construction especially interesting, is that the anticanonical system is empty: because of this, our mirrors are not covered by any other construction known to us. We discuss connections to the work of Beukers, Cohen and Mellit on hypergeometric functions.

math.AG↗