Title: Instanton and Ulrich bundles on cubics

Abstract: 

Lecture 1. Existence problems for Ulrich and instanton bundles 

§1. Ulrich rank and Ulrich complexity problems 

We will begin by recalling the basic features of Ulrich bundles and the Ulrich rank/complexity problems: determining the smallest possible rank of an Ulrich sheaf on a given polarized variety and describing all ranks that can occur. We will then review examples where these questions are well understood, especially in low dimension or for classical varieties such as Veronese embeddings. 

§2. Instanton bundles 

We will review the notion of instanton bundles and give an overview of their existence on several classes of varieties, notably Fano threefolds and hypersurfaces, highlighting their relation to the Ulrich rank problem. For rank-two instantons on hypersurfaces, we will recall the link with generalized Pfaffian representations via Steiner bundles. Along the way, we will introduce Coble cubics as Pfaffian representations in the framework of Vinberg’s θ-groups. 

Lecture 2. Ulrich and instanton bundles on cubic fourfolds 

§1. Sheaves on cubic fourfolds and the Kuznetsov category 

We will introduce the Kuznetsov category of a cubic fourfold and discuss its relationship with instantons, their acyclic extensions, and Ulrich bundles. We will briefly mention the associated deformation/obstruction theory and some hyperkähler moduli spaces arising from this category. We will then examine properties of Ulrich and instanton bundles in this setting, in particular their unobstructedness and the deformation of instantons into Ulrich bundles. 

§2. Ulrich and instanton sheaves on very general vs. special fourfolds 

We will then consider cubic fourfolds of different types, focusing on very general fourfolds and those lying in specific Hassett divisors. Finally, we will return to the Ulrich rank/complexity problem for cubic fourfolds. These lectures are mainly based on the recent preprint arXiv:2511.14470 with F. Galluzzi and G. Casnati.

Title: An introduction to vector bundles on algebraic varieties (introductory lectures for graduate students)

1. Vector bundles on algebraic curves 

2. Vector bundles on algebraic surfaces 

3. Vector bundles on higher dimensional algebraic varieties 

Abstract: 

1. In the first lecture, we will briefly review some basic notions and results about vector bundles on algebraic curves, e.g., slope stability, (semi-)stable bundle, classification of bundles on the projective line, elliptic curves and moduli space of vector bundles on higher genus curves, etc. 

2. In the second lecture, we will briefly review some basic notions and results about vector bundles on algebraic surfaces, e.g., ACM bundles, Ulrich bundles, Gieseker stability, (semi-)stable sheaves, moduli space of vector bundles on K3 surfaces, etc. 

3. In the third lecture, we will briefly review some basic notions and results about vector bundles on higher dimensional algebraic varieties, e.g., instanton bundles, Bridgeland stability, (semi-)stable complexes, moduli space of vector bundles on some Fano 3-folds, etc.

Title: Three lectures on vector bundles 

1. Lagrangian subspaces of the moduli space of simple sheaves on K3 surfaces. Applications 

2. Homogeneous and uniform vector bundles on projective spaces 

3. Ulrich bundles on general surfaces in a projective space 

Abstract: 

1. In the first lecture, we will present a new and useful method to construct isotropic and Lagrangian subspaces of the moduli space of simple sheaves on K3 surfaces.

2. In the second lecture, we will address the long standing open conjecture: Any uniform vector bundle of rank r<2n on a projective space of dimension n is homogeneous.

3. In the third lecture, we will discuss the problem of determining the pairs (r, d) such that a general surface of degree d in the 3-dimensional projective space supports an Ulrich bundle of rank r.

Title: Many facets of parabolic bundles

Abstract: A parabolic bundle is one of the simplest kinds of decorated vector bundles. It is a vector bundle with a collection of flags of subspaces on a fixed set of special points. In this talk, we review the basics of parabolic bundles, including several equivalent ways to understand them, the geometry of their moduli spaces, and some connections with other branches of mathematics, such as representation theory and conformal field theory. We also highlight some questions.

Title: Introduction to the theory of Ulrich, instanton and logarithmic bundles

Abstract: 

Lecture I (Ulrich bundles): Given a polarized projective variety (X, H), Ulrich bundles on it are the vector bundles with the simplest table of cohomology groups. In this lecture we are going to introduce the different approaches to the definition of Ulrich bundles, present different methods of construction and state the leading questions related to their existence. 

Lecture II (Instanton bundles): Instanton bundles on P^3 were defined as the algebraic geometric counterpart to SU(2)-connections with self-dual curvature on the real sphere S^4. Their study prompted the development of many techniques that have become central in algebraic geometry (monads, loci of jumping lines...). Recently its definition has been extended to arbitrary polarized projective varieties. In this lecture we are going to review the main points of this theory. 

Lecture III (Logarithmic sheaves on smooth projective varieties): Given a smooth projective variety X and a reduced divisor D on it,  the sheaf of logarithmic differential forms $\Omega^1_X(log D)$ is defined as the sheaf of one-forms having logarithmic poles along D. The sheaf was studied extensively by Deligne and Saito in connection with Hodge theory. On the other hand, this sheaf, or equivalently its dual sheaf $\mathcal{T}_X(- log D)$ of vector fields tangent along D, has been investigated from the point of view of stability theory, as well as related to Torelli problems. In this lecture we will introduce this objects and, time permitting, some generations.

Title: Hyperelliptic curves and Ulrich sheaves on the complete intersection of two quadrics

Subtitle 1: Vector bundles on hyperelliptic curves, Koszul algebras and BGG 

Subtitle 2: Clifford algebra, Morita equivalence and Reid's theorem 

Subtitle 3: Tate resolutions of O_X-modules and Ulrich bundles 

Abstract: Using the connection between hyperelliptic curves, Clifford algebras, and smooth complete intersections X of two quadrics, we describe Ulrich bundles on X and construct some of minimal possible rank. 

We will start out describing vector bundles on hyperelliptic curves via matrix factoriztions. The Jacobian of the hyperelliptic curve associated to X can be identified with the space of maximal isotropic subspaces of the pencil by Reid's theorem. We give a proof of Reid's theorem and apply these techniques to describe Ulrich bundles on X. Finally we will construct Ulrich bundles of smallest possible rank on X.

Title: Deformation of tangent bundles of projective hypersurfaces

Abstract: We provide a concrete birational model of the moduli space of sheaves on a projective hypersurface which contains the tangent bundle. It turns out to be an affine variety modeled after an irreducible representation associated to the ambient projective space. This is a joint work with Kiryong Chung and Jun-Muk Hwang.

Title: Ulrich bundles on Fano threefolds V22

Abstract: Quiver representation attracts a lot of interest as they provide convenient tool for studying the moduli problems. In this talk, I will introduce an explicit description of the moduli space of stable Ulrich bundles on Fano threefolds V22 (Picard number 1), obtained by identifying it with a suitable moduli space of representations of quiver with relations. This talk is based on joint work with Kyoung-Seog Lee and Kyeong-Dong Park.

Title: Irreducible components of rank 2 Brill-Noether loci on $\nu$-gonal curves

Abstract: We study the geometry of rank 2 Brill-Noether loci inside the moduli space $U_C^s(2,d)$ of stable vector bundles on a smooth projective curve $C$ of degree $d$. Focusing on a general $\nu$-gonal curve of genus $g$, we analyze loci parametrizing bundles with a prescribed number $k$ of global sections in the range $2g-2 \le d \le 4g-4$. Our attention is devoted to Brill-Noether loci of speciality two and three, corresponding to the cases $k=d-2g+2+i$ with $i=2, 3$. We classify the irreducible components of these loci and determine their dimensions. Each component is shown to be generically smooth, and its geometric properties are explicitly described. In particular, we distinguish regular and superabundant components via birational invariants. Our approach combines extension theory, moduli space techniques, and recent advances in the Brill-Noether theory of line bundles on $\nu$-gonal curves. These results extend earlier work of Sundaram and Teixidor i Bigas and provide a near-complete description of rank-2 Brill-Noether loci of speciality two and three in the $\nu$-gonal setting. This contributes to the broader understanding of higher-rank Brill-Noether theory on special curves. 

This is joint work with Professor Flaminio Flamini (University of Rome, Tor Vergata) and Professor Seonja Kim (Chungwoon university). 

Title: Arithmetic Cohen–Macaulay Toric Vector Bundles on the Veronese Surface

Abstract: We investigate toric vector bundles on the Veronese surface that are arithmetically Cohen–Macaulay. Using Klyachko’s filtration description, we can translate the aCM condition into explicit combinatorial constraints on the associated filtrations. Interpreting the filtration data via quiver representations, the aCM condition is determined by whether the quiver representation formed by vector spaces with filtration indices satisfying prescribed constraints contains a direct summand of a given dimension vector. As an application, we count non-split rank two aCM toric bundles over the Veronese surface, and show that in arbitrary rank every aCM toric bundle decomposes as a direct sum of toric bundles of rank at most two.

Title: Resolution of two rational maps from the space of plane cubics

Abstract: The family of plane cubic curves is parametrized by the projective space of dimension nine. The assignment to a plane cubic curve its Hessian cubic is a 3-1 map. The Hessian cubic comes with a non-trivial 2-torsion divisor class, and it allows one to represent the Hessian as the symmetric determinant of a net of cubics. This defines a birational map from the projective space to the Grassmannian 𝐺(3,6). We use the notion of net logarithmic sheaves to describe this picture and resolve the aforementioned two rational maps. This is a joint work with Simone Marchesi and Joan Pons-Llopis.

Title: Moduli Spaces of Sheaves on Complete Intersections via Net Logarithmic Tangent Sheaves

Abstract: A subspace of the moduli space of semistable torsion-free sheaves on a smooth projective variety with fixed Chern classes can be realized as a locus of logarithmic sheaves associated with an arrangement of divisors on the variety. In this talk, I will introduce a method to describe the moduli space of stable sheaves on complete intersections via a generalized logarithmic tangent sheaf, termed the net logarithmic tangent sheaf. This approach allows us to describe non-generic boundary points that are not realized as logarithmic sheaves in the classical sense. This talk is based on joint work with Sukmoon Huh.

Title: Ulrich bundles on the intersection of two quadrics

Abstract: Ulrich bundles are vector bundles with maximal cohomological vanishing, which have been studied in various contexts of algebraic geometry and homological algebra. While their existence is conjectured for all projective varieties, general constructions are still limited. In this talk, I will discuss work in progress with Kyoung-Seog Lee and Han-Bom Moon on the case of intersections of two quadrics.

Title: Moduli spaces of rational curves in moduli spaces of vector bundles over curves

Abstract: Let M be the moduli space of stable vector bundles of rank r and degree d over a smooth projective curve of genus >1, where r and d are coprime. When r>1, the moduli space R of rational curves in M of fixed degree c is non-compact for c>1. In this talk, I will discuss the birational geometry of R and compare various moduli theoretic compactifications by Hilbert scheme, stable maps and quasi-maps. Of particular interest is the case of conics in M with r=2 where the Hecke correspondence plays a key role. I will compare the moduli theoretic compactifications with those birational models from geometric invariant theory. Based on an ongoing project with Sanghyeon Lee.

Title: Bigness of tangent bundles of projective bundles

Abstract: As in Mori’s solution to Hartshorne’s conjecture, as well as in the Campana–Peternell conjecture, the positivity of tangent bundles imposes restrictions on the geometry of the underlying variety. Recently, there has been progress on the problem of classifying varieties whose tangent bundles are big. In this direction, I will introduce basic properties related to the bigness of tangent bundles and then discuss several approaches that can be applied to this problem in the case of projective bundles.

Title: Resonance, Syzygies, and Rank-3 Ulrich bundles on V_5

Abstract: An Ulrich bundle E on an n-dimensional projective variety (X, O(1)) is a vector bundle whose module of twisted global sections is a maximal Cohen-Macaulay module having the maximal number of generators in degree 0. It was once studied by commutative algebraists, but after Eisenbud and Schreyer introduced its geometric viewpoint, many people discovered several important applications in wide areas of mathematics. In this pioneering paper they asked whether every projective variety supports an Ulrich sheaf, and if yes, then what is the smallest possible rank. Thanks to a number of studies, the answer for the both question is now well-understood for del Pezzo threefolds. In particular, a del Pezzo threefold V_d of (degree d≥3) has an Ulrich bundle of rank r for every r at least 2. The Hartshorne-Serre correspondence translates the existence of rank-3 Ulrich bundle into the existence of an ACM curve C in V_d of genus g=2d+4 and degree 3d+3. The construction of rank-3 Ulrich bundle on a (general) cubic threefold is first suggested by Arrondo-Costa and then by Geiss-Schreyer, by showing that a "random" curve of given genus and degree lies in a cubic threefold and satisfies the whole conditions we needed. We discuss how this problem is related to the unirationality of the Hurwitz space H(k, 2g+2k-2) and the moduli of curves M_g. An analogous construction works for d=4, however, for d=5 a general curve of genus 14 and degree 18 in P^6 is not contained in V_5, observed in a work of Verra on the unirationality of M_14, and also pointed out recently by Ciliberto-Flamini-Knutsen. We characterize geometric conditions when does such a curve can be embedded into V_5 using the vanishing resonance. This is a joint work with Marian Aprodu.

Title: Moduli of equivariant symplectic/orthogonal bundles over toric varieties and their Euler characteristics

Abstract: Klyachko classified equivariant vector bundles over toric varieties and described them in terms of discrete data and elements in products of Grassmannians. Using this result, Kool related the GIT stability of products of Grassmannians under the GL(r)-action to the stability of vector bundles; furthermore, he described the torus fixed locus of the moduli space of sheaves in terms of these GIT quotients. As an application, he computed the Euler characteristics of the moduli space of slope-stable sheaves over smooth projective toric surfaces. 

Recently, Kavesh and Manon generalized Klyachko's classification to equivariant G-bundles. Building on this generalization, we describe the torus fixed locus of the moduli space of G-sheaves for G=Sp(2n), O(r), and SO(r). Moreover, we characterize these fixed loci in terms of GIT quotients of isotropic Grassmannians under the G-action. We also compute the Euler characteristics of these moduli spaces when the base space is a smooth projective toric surface, such as P^2. We will also compare the cases G=Sp(4) and SO(5), and will suggest some relationship between cases G=Sp(2n) and G=SO(2n+1).

Title: Geometry of parabolic $GL_n$-Higgs bundles

Abstract: The moduli space of Higgs bundles on a smooth curve carries rich geometry, including its hyperkähler structure (the non-abelian Hodge correspondence) and its algebraically integrable system structure given by the Hitchin fibration (the spectral correspondence). These structures naturally extend to the setting of parabolic Higgs bundles. In this talk, I will discuss how these structures extend to moduli spaces of parabolic Higgs bundles, by introducing $\xi$-parabolic Higgs bundles (diagonally parabolic Higgs bundles) and their moduli as building blocks. This is joint work with Jia Choon Lee.

Title: Positivity of the (co)tangent bundle of smooth projective varieties

Abstract: The positivity of (co)tangent bundle has profound implications for its geometry, as various conditions like ampleness, bigness, semi-ampleness, nefness, or pseudo-effectivity restrict its structure. In this surveying talk, I will discuss some geometric restrictions and applications on smooth projective manifolds due to the positivity of the (co)tangent bundle.

Title: Virtual Poincar\'e polynomial of moduli space of semistable sheaves of rank two on reducible curves

Abstract: We give an explicit description of the moduli space of rank-two semistable sheaves on a stable curve $C$ obtained by gluing two smooth curves at a point. We show that this moduli space is irreducible and birational to a projective bundle over the product of the moduli spaces of stable vector bundles on each component curve. This result holds for any choice of polarization. As an application, we compute the virtual Poincar\'e polynomial of this moduli space and describe how it relates to the corresponding polynomial in the smooth case.

Title: Curious Chern symmetry of moduli space of stable bundles on curves

Abstract: The cohomology ring of the moduli space of bundles on curves has long been studied, leading to an almost complete understanding of its structure. This cohomology ring admits a refined structure called the Chern filtration. Numerical experiments suggest that this refinement exhibits a certain symmetry. In this talk, I will explain a conjectural symmetry of the Chern filtration of the moduli space of bundles on curves, motivated by the curious Hard Lefschetz symmetry of character varieties. This is a joint work with M. Moreira and W. Pi.

Title: A birational geometry of the moduli space of $\mathcal{O}$-twisted rank 2 Hitchin pairs on a smooth curve 

Abstract: In 1999, Ch. Okonek, A. Schmitt and A. Teleman constructed a chain of $\mathbb{C}^*$-flips connecting to the moduli space of torsion free coherent sheaves on a smooth projective scheme via a variation of moduli of framed modules. In 2000, A. Schmitt constructed a chain of $\mathbb{C}^*$-flips connecting to the moduli space of Hitchin pairs on a smooth projective scheme via a variation of moduli of framed Hitchin pairs. Let $X$ be a smooth curve. In this work, we show that there exists a forgetful diagram from the chain of $\mathbb{C}^*$-flips of the moduli spaces of $\mathcal{O}_{X}$-twisted rank 2 framed Hitchin pairs on $X$ with Higgs fields compatible with framings to the chain of $\mathbb{C}^*$-flips of the moduli spaces of rank 2 framed modules on $X$. This is a joint work with YongJoo Shin.