arXiv subjects

Daniele Agostini

Publications and source records attributed to Daniele Agostini.

At least 19 recordsLinked to original sources

arXiv2026

Ulrich sheaves and determinantal representations for higher secant varieties of curves

We show that higher secant varieties of smooth projective curves have symmetric admissible determinantal representations with symmetric Ulrich sheaves of rank one if the embedding is sufficiently ample. For secant varieties of real curves, we give conditions for the representing matrices to be positive definite. This allows us, under mild (conjecturally vacuous) conditions, to represent the convex hull of the curve as a spectrahedron whenever it is a hyperbolicity cone. For secant varieties of rational normal curves, we derive very explicit representations in terms of Littlewood--Richardson coefficients. One key tool that we use are the higher Szegő kernels and the higher Scorza correspondences associated to a non-effective theta characteristic on the curve.

math.AG↗

arXiv2026

Evaluation of Large Language Models' educational feedback in Higher Education: potential, limitations and implications for educational practice

The importance of managing feedback practices in higher education has been widely recognised, as they play a crucial role in enhancing teaching, learning, and assessment processes. In today's educational landscape, feedback practices are increasingly influenced by technological advancements, particularly artificial intelligence (AI). Understanding the impact of AI on feedback generation is essential for identifying its potential benefits and establishing effective implementation strategies. This study examines how AI-generated feedback supports student learning using a well-established analytical framework. Specifically, feedback produced by different Large Language Models (LLMs) was assessed in relation to student-designed projects within a training course on inclusive teaching and learning. The evaluation process involved providing seven LLMs with a structured rubric, developed by the university instructor, which defined specific criteria and performance levels. The LLMs were tasked with generating both quantitative assessments and qualitative feedback based on this rubric. The AI-generated feedback was then analysed using Hughes, Smith, and Creese's framework to evaluate its structure and effectiveness in fostering formative learning experiences. Overall, these findings indicate that LLMs can generate well-structured feedback and hold great potential as a sustainable and meaningful feedback tool, provided they are guided by clear contextual information and a well-defined instructions that will be explored further in the conclusions.

cs.CY↗

arXiv2025

Fixed divisors on hyperkähler manifolds

Let $X$ be a hyperkähler manifold, and let $A$ be a nef and big divisor on $X$. We show that the fixed part of the linear system $|A|$ is reduced and as a consequence $|2A|$ is mobile. If $X$ has dimension four we also show that if the fixed part of $|A|$ is not empty, the mobile part induces a (rational) Lagrangian fibration.

math.AG↗

arXiv2025

Determinantal ideals of secant varieties

Using Hilbert schemes of points, we establish a number of results for a smooth projective variety $X$ in a sufficiently ample embedding. If $X$ is a curve or a surface, we show that the ideals of higher secant varieties are determinantally presented, and we prove the same for the first secant variety if $X$ has arbitrary dimension. This completely settles a conjecture of Eisenbud-Koh-Stillman for curves and partially resolves a conjecture of Sidman-Smith in higher dimensions. If $X$ is a curve or a surface we also prove that the corresponding embedding of the Hilbert scheme of points $X^{[d]}$ into the Grassmannian is projectively normal. Finally, if $X$ is an arbitrary projective scheme in a sufficiently ample embedding, then we demonstrate that its homogeneous ideal is generated by quadrics of rank three, confirming a conjecture of Han-Lee-Moon-Park. Along the way, we check that the Hilbert scheme of three points on a smooth variety is the blow-up of the symmetric product along the big diagonal.

math.AG↗

arXiv2025

On the irrationality of moduli spaces of projective hyperkähler manifolds

The aim of this paper is to estimate the irrationality of moduli spaces of hyperkähler manifolds of types K3$^{[n]}$, Kum$_{n}$, OG6, and OG10. We prove that the degrees of irrationality of these moduli spaces are bounded from above by a universal polynomial in the dimension and degree of the manifolds they parametrize. We also give a polynomial bound for the degrees of irrationality of moduli spaces of $(1,d)$-polarized abelian surfaces.

math.AG↗

arXiv2025

Coble duality for Jacobian Kummer fourfolds

We study projective models of generalized Kummer fourfolds via O'Grady's theta groups and the classical Coble cubic. More precisely, we establish a duality between two singular models of the generalized Kummer fourfold of a Jacobian abelian surface. We also give projective models for singular Jacobian Kummer varieties of arbitrary dimension. Along the way, we also construct a first non-natural involution on the Hilbert square of a Jacobian surface. In the appendix, we study singularities of secants of arbitrary varieties at identifiable points, following Choi, Lacini, Park and Sheridan.

math.AG↗

arXiv2025

Ulrich sheaves, the arithmetic writhe and algebraic isotopies of space curves

We establish a connection between the theory of Ulrich sheaves and $\mathbb{A}^1$-homotopy theory. For instance, we prove that the $\mathbb{A}^1$-degree of a morphism between projective varieties, that is relatively oriented by an Ulrich sheaf, is constant on the target even when it is not $\mathbb{A}^1$-chain connected or $\mathbb{A}^1$-connected. Further if an embedded projective variety is the support of a symmetric Ulrich sheaf of rank one, the $\mathbb{A}^1$-degree of all its linear projections can be read off in an explicit way from the free resolution of the Ulrich sheaf. Finally, we construct an Ulrich sheaf on the secant variety of a curve and use this to define an arithmetic version of Viro's encomplexed writhe for curves in $\mathbb{P}^3$. This can be considered to be an arithmetic analogue of a knot invariant. Namely, we define a notion of algebraic isotopy under which the arithmetic writhe is invariant. For rational curves of degree at most four in $\mathbb{P}^3$ we obtain a complete classification up to algebraic isotopies.

math.AG↗

arXiv2024

Points on Rational Normal Curves and the ABCT Variety

The ABCT variety is defined as the closure of the image of $G(2,n)$ under the Veronese map. We realize the ABCT variety $V(3,n)$ as the determinantal variety of a vector bundle morphism. We use this to give a recursive formula for the fundamental class of $V(3,n)$. As an application, we show that special Schubert coefficients of this class are given by Eulerian numbers, matching a formula by Cachazo-He-Yuan. On the way to this, we prove that the variety of configuration of points on a common divisor on a smooth variety is reduced and irreducible, generalizing a result of Caminata-Moon-Schaffler.

math.AG↗

arXiv2024

Plane quartics and heptagons

Every polygon with n vertices in the complex projective plane is naturally associated with its adjoint curve of degree n-3. Hence the adjoint of a heptagon is a plane quartic. We prove that a general plane quartic is the adjoint of exactly 864 distinct complex heptagons. This number had been numerically computed by Kohn et al. We use intersection theory and the Scorza correspondence for quartics to show that 864 is an upper bound, complemented by a lower bound obtained through explicit analysis of the famous Klein quartic.

math.AG↗

arXiv2024

On the Prym map for cyclic covers of genus two curves

The Prym map assigns to each covering of curves a polarized abelian variety. In the case of unramified cyclic covers of curves of genus two, we show that the Prym map is ramified precisely on the locus of bielliptic covers. The key observation is that we can naturally associate to such a cover an abelian surface with a cyclic polarization, and then the codifferential of the Prym map can be interpreted in terms of multiplication of sections on the abelian surface. Furthermore, we give a different proof of a result by Ramanan that a genus two cyclic cover of degree sufficiently high is never hyperelliptic.

math.AG↗

arXiv2024

Vector Spaces of Generalized Euler Integrals

We study vector spaces associated to a family of generalized Euler integrals. Their dimension is given by the Euler characteristic of a very affine variety. Motivated by Feynman integrals from particle physics, this has been investigated using tools from homological algebra and the theory of $D$-modules. We present an overview and uncover new relations between these approaches. We also provide new algorithmic tools.

math.AG↗

arXiv2023

Recovery of Plane Curves from Branch Points

We recover plane curves from their branch points under projection onto a line. Our focus lies on cubics and quartics. These have 6 and 12 branch points respectively. The plane Hurwitz numbers 40 and 120 count the orbits of solutions. We determine the numbers of real solutions, and we present exact algorithms for recovery. Our approach relies on 150 years of beautiful algebraic geometry, from Clebsch to Vakil and beyond.

math.AG↗

arXiv2022

Likelihood Degenerations

Computing all critical points of a monomial on a very affine variety is a fundamental task in algebraic statistics, particle physics and other fields. The number of critical points is known as the maximum likelihood (ML) degree. When the variety is smooth, it coincides with the Euler characteristic. We introduce degeneration techniques that are inspired by the soft limits in CEGM theory, and we answer several questions raised in the physics literature. These pertain to bounded regions in discriminantal arrangements and to moduli spaces of point configurations. We present theory and practise, connecting complex geometry, tropical combinatorics, and numerical nonlinear algebra.

math.AG↗

arXiv2022

On the irrationality of moduli spaces of K3 surfaces

We study how the degrees of irrationality of moduli spaces of polarized K3 surfaces grow with respect to the genus $g$. We prove that the growth is bounded by a polynomial function of degree $14+\varepsilon$ for any $\varepsilon>0$ and, for three sets of infinitely many genera, the bounds can be refined to polynomials of degree $10$. The main ingredients in our proof are the modularity of the generating series of Heegner divisors due to Borcherds and its generalization to higher codimensions due to Kudla, Millson, Zhang, Bruinier, and Westerholt-Raum. For special genera, the proof is also built upon the existence of K3 surfaces associated Hodge theoretically with certain cubic fourfolds, Gushel-Mukai fourfolds, and hyperkähler fourfolds.

math.AG↗

arXiv2021

On Algebraic Theta Divisors and Rational Solutions of the KP Equation

In this paper we classify the singular curves whose theta divisors in their generalized Jacobians are algebraic, meaning that they are cut out by polynomial analogs of theta functions. We also determine the degree of an algebraic theta divisor in terms of the singularities of the curve. Furthermore, we show a precise relation between such algebraic theta functions and the corresponding tau functions for the KP hierarchy.

math.AG↗

arXiv2021

The Martens-Mumford Theorem and the Green-Lazarsfeld Secant Conjecture

The Green-Lazarsfeld secant conjecture predicts that the syzygies of a curve of sufficiently high degree are controlled by its special secants. We prove this conjecture for all curves of Clifford index at least two and not bielliptic and for all line bundles of a certain degree. Our proof is based on a classic result of Martens and Mumford on Brill-Noether varieties and on a simple vanishing criterion that comes from the interpretation of syzygies through symmetric products of curves.

math.AG↗

arXiv2021

KP Solitons from Tropical Limits

We study solutions to the Kadomtsev-Petviashvili equation whose underlying algebraic curves undergo tropical degenerations. Riemann's theta function becomes a finite exponential sum that is supported on a Delaunay polytope. We introduce the Hirota variety which parametrizes all tau functions arising from such a sum. We compute tau functions from points on the Sato Grassmannian that represent Riemann-Roch spaces and we present an algorithm that finds a soliton solution from a rational nodal curve.

math.AG↗

arXiv2021

Pencils on surfaces with normal crossings and the Kodaira dimension of $\overline{\mathcal{M}}_{g,n}$

We study smoothing of pencils of curves on surfaces with normal crossings. As a consequence we show that the canonical divisor of $\overline{\mathcal{M}}_{g,n}$ is not pseudo-effective in some range, implying that $\overline{\mathcal{M}}_{12,6},\overline{\mathcal{M}}_{12,7},\overline{\mathcal{M}}_{13,4}$ and $\overline{\mathcal{M}}_{14,3}$ are uniruled. We provide upper bounds for the Kodaira dimension of $\overline{\mathcal{M}}_{12,8}$ and $\overline{\mathcal{M}}_{16}$. We also show that the moduli of $(4g+5)$-pointed hyperelliptic curves $\mathcal{H}_{g,4g+5}$ is uniruled. Together with a recent result of Schwarz, this concludes the Kodaira classification for moduli of pointed hyperelliptic curves.

math.AG↗