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Claudio Fontanari

Publications and source records attributed to Claudio Fontanari.

At least 19 recordsLinked to original sources
arXiv2026

On the rational cohomology of moduli spaces of Prym curves

We investigate the low degree rational cohomology groups of the moduli space of twisted Prym curves ${\overline{Pr}_{g,n}^{\hspace{0.05cm}(m_1, \ldots, m_n)}} $, where the integer twists $0\leq m_i\leq 1$ have even sum over $i$. We prove that these groups vanish in odd degree $\leq3$ and that the group in degree $2$ is algebraic. In particular, the results cover the classical moduli spaces of Prym curves and Prym curves with simple ramifications.

math.AG↗
arXiv2026

On the Riemann-Roch formula: old and new

The Riemann-Roch formula is a cornerstone in the classical theory of algebraic curves. Here we present a novel approach to its proof, by answering a question posed in 2007 by Matthew Baker and Serguei Norine.

math.AG↗
arXiv2025

IMProofBench: Benchmarking AI on Research-Level Mathematical Proof Generation

As the mathematical capabilities of large language models (LLMs) improve, it becomes increasingly important to evaluate their performance on research-level tasks at the frontier of mathematical knowledge. However, existing benchmarks are limited, as they focus solely on final-answer questions or high-school competition problems. To address this gap, we introduce IMProofBench, a private benchmark consisting of 77 peer-reviewed problems developed by expert mathematicians. Each problem requires a detailed proof and is paired with subproblems that have final answers, supporting both an evaluation by human experts and a large-scale quantitative analysis through automated grading. Furthermore, unlike prior benchmarks, the evaluation setup simulates a realistic research environment: models operate in an agentic framework with tools like web search for literature review and mathematical software such as SageMath. Our results show that current LLMs can already solve a significant percentage of research-level questions. IMProofBench will continue to evolve as a dynamic benchmark in collaboration with the mathematical community, ensuring its relevance for evaluating the next generation of LLMs.

cs.CL↗
arXiv2024

Terracini loci and a codimension one Alexander-Hirschowitz theorem

The Terracini locus $\mathbb{T}(n, d; x)$ is the locus of all finite subsets $S$ of $ \mathbb{P}^n$ of cardinality $x$ such that $\langle S \rangle = \mathbb{P}^n$, $h^0(\mathcal{I}_{2S}(d)) > 0$, and $h^1(\mathcal{I}_{2S}(d)) > 0$. The celebrated Alexander-Hirschowitz Theorem classifies the triples $(n,d,x)$ for which $\dim\mathbb{T}(n, d; x)=xn$. Here we fully characterize the next step in the case $n=2$, namely, we prove that $\mathbb{T}(2,d;x)$ has at least one irreducible component of dimension $2x-1$ if and only if either $(d,x)\in\{(4,4),(4,6),$ $(5,6),(5,7),$ $(6,9),(6,10)\}$, or $d\ge 7$, $d\equiv 1,2 \pmod{3}$ and $x=(d+2)(d+1)/6$.

math.AG↗
arXiv2024

Nonvanishing and Abundance for cones of movable divisors

Let $\overline{\mathrm{Mov}}^k(X)$ be the closure of the cone $\mathrm{Mov}^k(X)$ generated by classes of effective divisors on a projective variety $X$ with stable base locus of codimension at least $k+1$. We propose a generalized version of the Log Nonvanishing Conjecture and of the Log Abundance Conjecture for a klt pair $(X,Δ)$, that is: if $K_X+Δ\in \overline{\mathrm{Mov}}^{k}(X)$, then $K_X+Δ\in \mathrm{Mov}^{k}(X)$. Moreover, we prove that if the Log Minimal Model Program, the Log Nonvanishing, and the Log Abundance hold, then so does our conjecture.

math.AG↗
arXiv2024

A question about generalized nonvanishing

Let $(X,Δ)$ be a projective, $\mathbb{Q}$-factorial log canonical pair and let $L$ be a pseudoeffective $\mathbb{Q}$-divisor on $X$ such that $K_X + Δ+ L$ is pseudoeffective. Is there an effective $\mathbb{Q}$-divisor $M$ on $X$ such that $K_X + Δ+ L$ is numerically equivalent to $M$? We are not aware of any counterexamples, but the answer is not completely clear even in the case of surfaces.

math.AG↗
arXiv2023

On varieties whose general surface section has negative Kodaira dimension

In this paper, inspired by work of Fano, Morin and Campana--Flenner, we give a full projective classification of (however singular) varieties of dimension 3 whose general hyperplane sections have negative Kodaira dimension, and we partly extend such a classification to varieties of dimension $n\geq 4$ whose general surface sections have negative Kodaira dimension. In particular we prove that a variety of dimension $n\geq 3$ whose general surface sections have negative Kodaira dimension is birationally equivalent to the product of a general surface section times $\p^{n-2}$ unless (possibly) if the variety is a cubic hypersurface.

math.AG↗
arXiv2023

A remark on generalized abundance for surfaces

Let $(X, Δ)$ be a projective klt pair of dimension $2$ and let $L$ be a nef $\mathbb{Q}$-divisor on $X$ such that $K_X + Δ+ L$ is nef. As a complement to the Generalized Abundance Conjecture by Lazić and Peternell, we prove that if $K_X + Δ$ and $L$ are not proportional modulo numerical equivalence, then $K_X + Δ+ L$ is semiample. An example due to Lazić shows that this is no longer true in any dimension $n \ge 3$.

math.AG↗
arXiv2023

Holomorphic differential forms on moduli spaces of stable curves

We prove that the space of holomorphic $p$-forms on the moduli space $\overline{\mathcal{M}}_{g,n}$ of stable curves of genus $g$ with $n$ marked points vanishes for $p=14, 16, 18$ unconditionally and also for $p=20$ under a natural assumption in the case $g=3$. This result is consistent with the Langlands program and it is obtained by applying the Arbarello-Cornalba inductive approach to the cohomology of moduli spaces.

math.AG↗
arXiv2023

A remark on the canonical degree of curves on smooth projective surface

The canonical degree $C.K_X$ of an integral curve on a smooth projective surface $X$ is conjecturally bounded from above by an expression of the form $A(g-1)+B$, where $g$ is the geometric genus of $C$ and $A$, $B$ are constants depending only on $X$. We prove that this conjecture holds with $A = -1$ under the assumptions $h^0(X, -K_X) = 0$ and $h^0(X, 2K_X + C) = 0$.

math.AG↗
arXiv2023

A remark on weighted Bounded Negativity for blow-ups of the projective plane

Motivated by the weighted Bounded Negativity Conjecture, we prove that all but finitely many reduced and irreducible curves $C$ on the blow-up of $\mathbb{P}^2$ at $n$ points satisfy the inequality $C^2 \ge \min \{-\frac{1}{12} n (C.L +27), -2 \}$, where $L$ is the pull-back of a line. This partially improves on some result by Laface and Pokora.

math.AG↗
arXiv2022

Objectivity and Rigor in Classical Italian Algebraic Geometry

The classification of algebraic surfaces by the Italian School of algebraic geometry is universally recognized as a breakthrough in 20th-century mathematics. The methods by which it was achieved do not, however, meet the modern standard of rigor and therefore appear dubious from a contemporary viewpoint. In this article, we offer a glimpse into the mathematical practice of the three leading exponents of the Italian School of algebraic geometry: Castelnuovo, Enriques, and Severi. We then bring into focus their distinctive conception of rigor and intuition. Unlike what is often assumed today, from their perspective, rigor is neither opposed to intuition nor understood as a unitary phenomenon - Enriques distinguishes between small-scale rigor and large-scale rigor and Severi between formal rigor and substantial rigor. Finally, we turn to the notion of mathematical objectivity. We draw from our case study in order to advance a multi-dimensional analysis of objectivity. Specifically, we suggest that various types of rigor may be associated with different conceptions of objectivity: namely objectivity as faithfulness to facts and objectivity as intersubjectivity.

math.HO↗
arXiv2022

Two letters by Guido Castelnuovo

In this expository paper we transcribe two letters by Guido Castelnuovo, one to Francesco Severi, the other to Beniamino Segre, and explain the contents of both, which basically focus on the quest for an algebraic proof of the equality between the analytic and the arithmetic irregularity and of the closedness of regular 1-forms on a complex, projective, algebraic surface. Such an algebraic proof has been found only in the 1980's by Deligne and Illusie.

math.AG↗
arXiv2022

On the existence of finite surjective parametrizations of affine surfaces

We investigate surjective parametrizations of rational algebraic varieties, in the vein of recent work by Jorge Caravantes, J. Rafael Sendra, David Sevilla, and Carlos Villarino. In particular, we show how to construct plenty of examples of affine surfaces $S$ not admitting a finite surjective morphism $f: \mathbb{A}^2 \to S$.

math.AG↗
arXiv2021

On the Splitting Principle of Beniamino Segre

We state and prove in modern terms a Splitting Principle first claimed by Beniamino Segre in 1938, which should be regarded as a strong form of the classical Principle of Connectedness.

math.AG↗