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Joan Pons-Llopis

Publications and source records attributed to Joan Pons-Llopis.

11 recordsLinked to original sources
arXiv2026

Ulrich Bundles on decomposable threefold scrolls over $\mathbb F_a$

Ulrich bundles provide profound insights into the underlying geometry and derived categories of projective varieties supporting them, yet their existence and modular properties remain sometimes largely obscure. In this paper we study the geometry and the moduli spaces of Ulrich bundles on broad families of decomposable threefold scrolls $X$ over Hirzebruch surfaces $\mathbb F_a$, proving that their Ulrich complexity is 1. Exploiting the double-scroll structure enjoyed by $X$, we also introduce a (geometric) involution acting on the set of classified Ulrich line bundles which, together with the natural one, shapes the study of higher-rank extensions and significantly streamlines their modular study. We moreover prove that some higher-rank extensions yield indecomposable Ulrich bundles whose existence is intrinsically $3$-dimensional, i.e. going beyond natural pullbacks from the base surfaces. In rank two, for any choice of the parameters involved, we provide a comprehensive description of associated modular irreducible components, determining their dimensions, their generic smoothness and the description of their birational structure. Ultimately, for noteworthy parameter cases, we focus on the Ulrich representation type of $X$ proving that it is Ulrich wild by the existence of generically smooth modular components of slope-stable Ulrich bundles of arbitrary rank $r$ and of dimension growing quadratically with $r$, which reveals the unbounded complexity of Ulrich modules supported on these threefolds.

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arXiv2025

't Hooft bundles on the complete flag threefold and moduli spaces of instantons

In this work we study the moduli spaces of instanton bundles on the flag twistor space $F:=F(0,1,2)$. We stratify them in terms of the minimal twist supporting global sections and we introduce the notion of (special) 't Hooft bundle on $F$. In particular we prove that there exist $μ$-stable 't Hooft bundles for each admissible charge $k$. We completely describe the geometric structure of the moduli space of (special) 't Hooft bundles for arbitrary charge $k$. Along the way to reach these goals, we describe the possible structures of multiple curves supported on some rational curves in $F$ as well as the family of del Pezzo surfaces realized as hyperplane sections of $F$. Finally we investigate the splitting behaviour of 't Hooft bundles when restricted to conics.

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arXiv2023

Generalized logarithmic sheaf on smooth projective surfaces

We define the notion of generalized logarithmic sheaves on a smooth projective surface, associated to a pair consisting of a reduced curve and some fixed points on it. We then set up the study of the Torelli property in this setting, focusing mostly in the case of the blow-up of the projective plane on a reduced set of points and, in particular, in the case of the cubic surface. We also study the stability property of generalized logarithmic sheaves as well as carrying out the description of their moduli spaces.

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arXiv2019

Instanton bundles on the flag variety F(0,1,2)

Instanton bundles on $\mathbb{P}^3$ have been at the core of the research in Algebraic Geometry during the last thirty years. Motivated by the recent extension of their definition to other Fano threefolds of Picard number one, we develop the theory of instanton bundles on the complete flag variety $F:=F(0,1,2)$ of point-lines on $\mathbb{P}^2$. After giving for them two different monadic presentations, we use it to show that the moduli space $MI_F(k)$ of instanton bundles of charge $k$ is a geometric GIT quotient and the open subspace $MI^s_F(k)\subset MI_F(k)$ of stable instanton bundles has a generically smooth component of dim $8k-3$. Finally we study their locus of jumping conics.

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arXiv2018

Ulrich bundles on three dimensional scrolls

In this paper we construct Ulrich bundles of low rank on three-dimensional scrolls (with respect to the tautological line bundle). We pay special attention to the four types of threefold scrolls in $\mathbb{P}^5$ which were classified in [Ott92].

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arXiv2018

aCM vector bundles on projective surfaces of nonnegative Kodaira dimension

In this paper we contribute to the construction of families of arithmetically Cohen-Macaulay (aCM) indecomposable vector bundles on a wide range of polarized surfaces $(X,\Oo_X(1))$ for $\Oo_X(1)$ an ample line bundle. In many cases, we show that for every positive integer $r$ there exists a family of indecomposable aCM vector bundles of rank $r$, depending roughly on $r$ parameters, and in particular they are of \emph{wild representation type}. We also introduce a general setting to study the complexity of a polarized variety $(X,\Oo_X(1))$ with respect to its category of aCM vector bundles. In many cases we construct indecomposable vector bundles on $X$ which are aCM for all ample line bundles on $X$.

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arXiv2018

ACM sheaves on the double plane

The goal of this paper is to start a study of aCM and Ulrich sheaves on non-integral projective varieties. We show that any aCM vector bundle of rank two on the double plane is a direct sum of line bundles. As a by-product, any aCM vector bundle of rank two on a sufficiently high dimensional quadric hypersurface also splits. We consider aCM and Ulrich vector bundles on a multiple hyperplanes and prove the existence of such bundles that do not split, if the multiple hyperplane is linearly embedded into a sufficiently high dimensional projective space. Then we restrict our attention to the double plane and give a classification of aCM sheaves of rank at most $3/2$ on the double plane and describe the family of isomorphism classes of them.

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arXiv2010

ACM bundles on del Pezzo surfaces

ACM rank 1 bundles on del Pezzo surfaces are classified in terms of the rational normal curves that they contain. A complete list of ACM line bundles is provided. Moreover, for any del Pezzo surface $X$ of degree less or equal than six and for any $n\geq 2$ we construct a family of dimension $\geq n-1$ of non-isomorphic simple ACM bundles of rank $n$ on $X$.

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