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Alex Massarenti

Publications and source records attributed to Alex Massarenti.

At least 19 recordsLinked to original sources
arXiv2026

Linear Independence of Polynomial Compositions and Identifiability of Deep Neural Networks

Motivated by theoretical problems in deep learning, we conjecture that post-composing a fixed number of pairwise distinct nonconstant polynomials with a generic polynomial of sufficiently large degree yields linearly independent polynomials. This generalizes Newman--Slater's theorem on powers of polynomials. We establish several cases of this conjecture and its origin-passing variant: We prove the result for two polynomials, and for an arbitrary number of polynomials when their degrees are bounded. Furthermore, we show how the conjecture implies a complete understanding of the identifiability (i.e., parameter symmetries) of deep fully connected neural network architectures with generic polynomial activation functions. In particular, for network architectures with layer-specific activations of increasing degree, our established versions of the conjecture fully characterize the set of parameters yielding the same end-to-end network function. As a special case, we fully resolve the identifiability of shallow polynomial networks.

math.AC↗
arXiv2026

Unirational Fano threefolds

We use the relative Abel--Jacobi construction of Cheltsov--Loginov--Orlov--Prokhorov to study unirational Fano threefolds. Our main result is a $50$-dimensional family of smooth unirational del Pezzo threefolds of degree one. We also construct explicit one-parameter families of terminal unirational sextic double solids and quartic threefolds of Picard rank one. Finally, we reinterpret the Roth--Iskovskikh unirationality construction for smooth double quadric threefolds in terms of a correspondence induced by the relative Abel--Jacobi map.

math.AG↗
arXiv2026

Mori theory and symplectic reduction for polygons and quivers

We investigate the birational geometry of moduli spaces of polygons and quivers, emphasizing the link between Mori theory and symplectic reduction. On the additive side we relate Mori models of $Bl_{n-1}\mathbb{P}^{n-3}$ with GIT quotients of configurations of points on $\mathbb{P}^1$, Hassett--GIT spaces, parabolic quotients with fixed trivial bundle, quiver moduli spaces, and additive polygon spaces. On the multiplicative side we relate Mori models of $Bl_n\mathbb{P}^{n-3}$ with moduli spaces of rank two parabolic bundles with trivial determinant, multiplicative polygons, multiplicative configurations, and compact multiplicative quiver quotients. In this way, the Mori chamber decompositions are matched with wall-crossing for the corresponding additive and multiplicative symplectic quotients. Finally, for the small models appearing in these two constructions, we compute the corresponding groups of algebraic automorphisms.

math.AG↗
arXiv2026

Log-Conformal Projective Manifolds

Let $(X,\Delta)$ be a smooth complex projective simple normal crossing pair of dimension $n\ge 3$ endowed with an everywhere nondegenerate logarithmic conformal tensor. If $K_X+\Delta$ is not nef, then exactly one of the following occurs: $\Delta=\varnothing$ and $X\simeq Q^n$; $X\simeq\mathbb{P}^n$ and $\Delta$ is a hyperplane; or $n=2m$ and $(X,\Delta)$ admits a $(K_X+\Delta)$-negative elementary contraction $\phi:X\to Y$ generated by minimal rational curves of conformal degree one. In the third case, if $\dim Y>0$, then for some $1\le s\le m$ one has $\dim Y=m-s+1$ and the geometric generic fiber is $(\mathbb{P}^{m+s-1},H_1+\cdots+H_s)$, where the $H_i$ are hyperplanes in general position; over a dense open subset, $\phi$ is a projective bundle and the horizontal boundary is the sum of $s$ relative hyperplanes. The case $s=1$ yields a rational maximal isotropic fibration with generic fiber $(\mathbb{P}^m,H)$; if $\dim Y=0$, then $\rho(X)=1$ and $-(K_X+\Delta)$ is ample. If $K_X+\Delta\equiv0$, then, under a Bochner extension principle, a restricted-holonomy condition, and trivial monodromy of the induced flat connection on $X\setminus\Delta$, a logarithmic conformal tensor with trivial conformal line bundle forces $X\setminus\Delta$ to be semi-abelian and $(X,\Delta)$ to be its toroidal compactification.

math.AG↗
arXiv2026

Compactifications of spaces of symmetric matrices and pointed Kontsevich spaces of isotropic Grassmannians

We study two closely related families of varieties arising from genus $0$ stable maps to the Lagrangian Grassmannian $\operatorname{LG}(n,2n)$. First, we construct the Kausz--type compactification $\mathcal {TL}_n$ of the space of symmetric matrices and give an explicit description of its birational geometry. Second, we realize $\mathcal {TL}_n$ as a general evaluation fiber in a Kontsevich space, and then exploit this modular interpretation to derive consequences for the birational geometry of the space of pointed conics $\overline{M}_{0,1}(\operatorname{LG}(n,2n),2)$. Analogous compactifications related to orthogonal Grassmannians are also presented.

math.AG↗
arXiv2026

Canonical blow-ups of Lagrangian and Orthogonal Grassmannians

Let $\mathbf{LG}(V\oplus V^*)$ and $\mathbf{OG}^+(V\oplus V^*)$ denote the Lagrangian and orthogonal Grassmannians endowed with the natural $\mathbb{G}_m$-actions, respectively. Thaddeus proved that over $\mathbb{C}$, the Hilbert quotients $\mathbf{LG}(V\oplus V^*)\!/\!/\mathbb{G}_m$ and $\mathbf{OG}^+(V\oplus V^*)\!/\!/\mathbb{G}_m$ are isomorphic to the wonderful compactifications of the spaces of symmetric and skew-symmetric matrices of maximal ranks, that is, the spaces of complete quadrics and complete skew-forms, respectively. In this paper, we construct the universal families of these Hilbert quotients by explicitly blowing up the corresponding isotropic Grassmannians, resulting in smooth toroidal compactifications of the spaces of symmetric and skew-symmetric matrices of maximal ranks (before projectivization), which have simple normal crossing boundary divisors and include the spaces of the complete bilinear forms among these divisors. Specifically, we prove that over any algebraically closed field, the universal families of these Hilbert quotients are smooth, and the Hilbert quotients themselves are isomorphic to the spaces of the complete bilinear forms. Over an algebraically closed field of characteristic zero, we prove that the universal families are weak Fano varieties with vanishing higher cohomology groups for their tangent bundles, and are therefore locally rigid. Furthermore, we show that these universal families naturally resolve the Landsberg-Manivel rational maps from projective spaces to isotropic Grassmannians.

math.AG↗
arXiv2025

On the unirationality of conic bundles with discriminant of degree eight

We study the unirationality of surface conic bundles $\pi\colon S\to\mathbb P^1$ over an arbitrary field $k$ with discriminant degree $d_S=8$, the first case beyond the del Pezzo range. We divide these surfaces in four families and produce explicit rational multisections via tangent constructions and Cremona transformations. Over $C_1$ fields we obtain Zariski dense loci of minimal, hence non $k$-rational, yet $k$-unirational conic bundles in each family; for one of the types we prove that the dense unirational locus is indeed Zariski open. Finally, we investigate the deformation theory of these conic bundles and how their unirationality behaves under specialization.

math.AG↗
arXiv2025

On K-stability of Fano's last Fanos

We study K-stability of smooth Fano threefolds of Picard rank $2$ and degree $22$ which can be obtained by blowing up a smooth complete intersection of two quadrics in $\mathbb{P}^5$ along a conic. We also describe the automorphism groups of these threefolds.

math.AG↗
arXiv2025

Algebraic growth of the Cremona group

We initiate the study of the ''algebraic growth'' of groups of automorphisms and birational transformations of algebraic varieties. Our main result concerns $\text{Bir}(\mathbb{P}^2)$, the Cremona group in $2$ variables. This group is the union, for all degrees $d\geq 1$, of the algebraic variety $\text{Bir}(\mathbb{P}^2)_d$ of birational transformations of the plane of degree $d$. Let $N_d$ denote the number of irreducible components of $\text{Bir}(\mathbb{P}^2)_d$. We describe the asymptotic growth of $N_d$ as $d$ goes to $+\infty$, showing that there are two constants $A$ and $B>0$ such that $$ A\sqrt{\ln(d)} \leq \ln \left(\ln \left(\sum_{e\leq d} N_e \right) \right) \leq B \sqrt{\ln(d)} $$ for all large enough degrees $d$. This growth type seems quite unusual and shows that computing the algebraic growth of $\text{Bir}(\mathbb{P}^2)$ is a challenging problem in general.

math.AG↗
arXiv2024

Mori dream singular $K3$ surfaces

We take a first step towards the classification of singular Mori dream $K3$ surfaces. We prove that if the Picard lattice of a singular $K3$ surface is Mori dream, then the surface is Mori dream. Moreover, we show that for singular $K3$ surfaces, of Picard rank two, being Mori dream is equivalent to contain two negative curves intersecting each other, and apply this result to study Mori dreamness of $K3$ surfaces with a singular point of type $A_n$.

math.AG↗
arXiv2024

Rational points on even dimensional Fermat cubics

We show that even dimensional Fermat cubic hypersurfaces are rational over any field of characteristic different from three by producing explicit rational parametrizations given by polynomials of low degree. As a byproduct of our rationality constructions we get estimates on the number of their rational points over a number field, and a class of quadro-cubic Cremona correspondences of even dimensional projective spaces.

math.AG↗
arXiv2023

Cox rings of blow-ups of multiprojective spaces

Let $X^{1,n}_r$ be the blow-up of $\mathbb{P}^1\times\mathbb{P}^n$ in $r$ general points. We describe the Mori cone of $X^{1,n}_r$ for $r\leq n+2$ and for $r = n+3$ when $n\leq 4$. Furthermore, we prove that $X^{1,n}_{n+1}$ is log Fano and give an explicit presentation for its Cox ring.

math.AG↗
arXiv2023

Birational geometry of Calabi-Yau pairs and 3-dimensional Cremona transformations

We develop a framework that allows one to describe the birational geometry of Calabi-Yau pairs $(X,D)$. After establishing some general results for Calabi-Yau pairs $(X,D)$ with mild singularities, we focus on the special case when $X=\mathbb{P}^3$ and $D\subset \mathbb{P}^3$ is a quartic surface. We investigate how the appearance of increasingly worse singularities on $D$ enriches the birational geometry of the pair $(\mathbb{P}^3, D)$.

math.AG↗
arXiv2022

Quartic and Quintic hypersurfaces with dense rational points

Let $X_4\subset\mathbb{P}^{n+1}$ be a quartic hypersurface of dimension $n\geq 4$ over an infinite field $k$. We show that if either $X_4$ contains a linear subspace $\Lambda$ of dimension $h\geq \max\{2,\dim(\Lambda\cap \text{Sing}(X_4))-2\}$ or has double points along a linear subspace of dimension $h\geq 3$, a smooth $k$-rational point and is otherwise general, then $X_4$ is unirational over $k$. This improves previous results by A. Predonzan and J. Harris, B. Mazur, R. Pandharipande for quartics. We also provide a density result for the $k$-rational points of quartic $3$-folds with a double plane over a number field, and several unirationality results for quintic hypersurfaces over a $C_r$ field.

math.AG↗
arXiv2022

Bronowski's conjecture and the identifiability of projective varieties

Let $X\subset\mathbb{P}^{hn+h-1}$ be an irreducible and non-degenerate variety of dimension $n$. The Bronowski's conjecture predicts that $X$ is $h$-identifiable if and only if the general $(h-1)$-tangential projection $\tau_{h-1}^X:X\dashrightarrow\mathbb{P}^n$ is birational. In this paper we provide counterexamples to this conjecture. Building on the ideas that led to the counterexamples we manage to prove an amended version of the Bronowski's conjecture for a wide class of varieties and to reduce the identifiability problem for projective varieties to their secant defectiveness.

math.AG↗