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Marian Aprodu

Publications and source records attributed to Marian Aprodu.

At least 19 recordsLinked to original sources
arXiv2025

The effective Chen ranks conjecture

Koszul modules and their associated resonance schemes are objects appearing in a variety of contexts in algebraic geometry, topology, and combinatorics. We present a proof of an effective version of the Chen ranks conjecture describing the Hilbert function of any Koszul module verifying natural conditions inspired by geometry. We give applications to hyperplane arrangements, describing in a uniform effective manner the Chen ranks of the fundamental group of the complement of every arrangement whose projective resonance is reduced. Finally, we formulate a sharp generic vanishing conjecture for Koszul modules and present a parallel between this statement and the Prym--Green Conjecture on syzygies of general Prym canonical curves.

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arXiv2025

Linear sections of Grassmannians and resonance of vector bundles

This work revolves around the question of whether a given resonance variety is associated with a vector bundle. We show the existence of a family of natural morphisms on a stratification of the resonance variety to a suitable family of a Quot scheme and provide some applications in the curve case. The existence of this family of morphisms represents an obstruction to affirmatively answering the main question. In addition, we study the resonance of restricted universal rank-two quotient bundles over transversal linear sections of the Grassmann varieties $\operatorname{Gr}_2(\mathbb{C}^n)$, with a special attention to low-dimensional Grassmannians. These bundles are among the most natural to consider in this context. The analysis for $\operatorname{Gr}_2(\mathbb{C}^6)$ shows that any resonance variety in $\mathbb{P}^5$ consisting of fourteen disjoint lines is the resonance of some bundle which appeared in the work of Mukai.

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arXiv2024

Resonance, syzygies, and rank-$3$ Ulrich bundles on the del Pezzo threefold $V_5$

We investigate a geometric criterion for a smooth curve $C$ of genus $14$ and degree $18$ to be described as the zero locus of sections in an Ulrich bundle of rank $3$ on a del Pezzo threefold $V_5 \subset \mathbb{P}^6$. The main challenge is to read off the Pfaffian quadrics defining $V_5$ from geometric structures of $C$. We find that this problem is related to the existence of a special rank-two vector bundle on $C$ with trivial resonance. It gives a description of the image of the rational map that appeared in a work of Ciliberto-Flamini-Knutsen, for the case of degree $5$ del Pezzo threefolds. From an explicit calculation of the Betti table of such a curve, we also deduce the uniqueness of the del Pezzo threefold containing a given curve.

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arXiv2024

The second syzygy schemes of curves of large degree

The present paper is a natural continuation of a previous work where we studied the second syzygy scheme of canonical curves. We find sufficient conditions ensuring that the second syzygy scheme of a genus--$g$ curve of degree at least $2g+2$ coincide with the curve. If the property $(N_2)$ is satisfied, the equality is ensured by a more general fact. If $(N_2)$ fails, then the analysis uses the known case of canonical curves.

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arXiv2024

Reduced resonance schemes and Chen ranks

The resonance varieties are cohomological invariants that are studied in a variety of topological, combinatorial, and geometric contexts. We discuss their scheme structure in a general algebraic setting and introduce various properties that ensure the reducedness of the associated projective resonance scheme. We prove an asymptotic formula for the Hilbert series of the associated Koszul module, then discuss applications to vector bundles on algebraic curves and to Chen ranks formulas for finitely generated groups, with special emphasis on Kähler and right-angled Artin groups.

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arXiv2024

Higher resonance schemes and Koszul modules of simplicial complexes

Each connected graded, graded-commutative algebra $A$ of finite type over a field $\Bbbk$ of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the (higher) Koszul modules of $A$. In this note, we investigate the geometry of the support loci of these modules, called the resonance schemes of the algebra. When $A=\Bbbk\langle Δ\rangle$ is the exterior Stanley-Reisner algebra associated to a finite simplicial complex $Δ$, we show that the resonance schemes are reduced. We also compute the Hilbert series of the Koszul modules and give bounds on the regularity and projective dimension of these graded modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group.

math.AC↗
arXiv2024

Sections of K3 surfaces with Picard number two and Mercat's conjecture

Farkas and Ortega found counterexamples to Mercat's conjecture by restricting to a hyperplane section $C$ some suitable rank-two vector bundles on a $K3$ surface whose Picard group is generated by $C$ and another very ample divisor. We prove that the same bundles produce other counterexamples by restriction to hypersurface sections $C_n\in|nC|$ for all $n\ge 2$. In the process, we compute the Clifford indices of the corresponding hypersurface sections $C_n$, noting their non-generic nature for $n\ge 2$. A key ingredient to prove the (semi)stability of the restricted bundles, is Green's Explicit $H^0$ Lemma. In what concerns the (semi)stability, although general restriction theorems as demonstrated by Flenner or Feyzbakhsh are applicable for sufficiently large, explicit values of $n$, our approach works for all $n\ge 2$. It is also worth noting that our proof deviates slightly from the one of Farkas-Ortega. Employing the same strategy leads to an enhancement of the main result of a paper of Sengupta.

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arXiv2023

Koszul modules with vanishing resonance in algebraic geometry

We discuss various applications of a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace in the second wedge product of a vector space. Previously Koszul modules of finite length have been used to give a proof of Green's Conjecture on syzygies of generic canonical curves. We now give applications to effective stabilization of cohomology of thickenings of algebraic varieties, divisors on moduli spaces of curves, enumerative geometry of curves on K3 surfaces and to skew-symmetric degeneracy loci. We also show that the stability of sufficiently positive rank 2 vector bundles on curves is governed by resonance.

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arXiv2023

Green's conjecture for general covers

We establish Green's syzygy conjecture for classes of covers of curves of higher Clifford dimension. These curves have an infinite number of minimal pencils, in particular they do not verify a well-known Brill-Noether theoretic sufficient condition that implies Green's conjecture. Secondly, we study syzygies of curves with a fixed point free involution and prove that sections of Nikulin surfaces of minimal Picard number 9, verify the classical Green Conjecture but fail the Prym-Green Conjecture on syzygies of Prym-canonical curves. This provides an explicit locus in the moduli space R_g where Green's Conjecture is known to hold.

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arXiv2022

On the monodromy map for the logarithmic differential systems

We study the monodromy map for logarithmic $\mathfrak g$-differential systems over an oriented surface $S_0$ of genus $g$, with $\mathfrak g$ being the Lie algebra of a complex reductive affine algebraic group $G$. These logarithmic $\mathfrak g$-differential systems are triples of the form $(X, D,Φ)$, where $(X, D) \in {\mathcal T}_{g,d}$ is an element of the Teichmüller space of complex structures on $S_0$ with $d \geq 1$ ordered marked points $D\subset S_0= X$ and $Φ$ is a logarithmic connection on the trivial holomorphic principal $G$-bundle $X \times G$ over $X$ whose polar part is contained in the divisor $D$. We prove that the monodromy map from the space of logarithmic $\mathfrak g$-differential systems to the character variety of $G$-representations of the fundamental group of $S_0\setminus D$ is an immersion at the generic point, in the following two cases: A) $g \geq 2$, $d \geq 1$, and $\dim_{\mathbb C}G \geq d+2$; B) $g=1$ and $\dim_{\mathbb C}G \geq d$. The above monodromy map is nowhere an immersion in the following two cases: 1) $g=0$ and $d \geq 4$; 2) $g\geq 1$ and $\dim_{\mathbb C}G < \frac{d+3g-3}{g}$. This extends to the logarithmic case the main results in \cite{CDHL}, \cite{BD} dealing with nonsingular holomorphic $\mathfrak g$-differential systems (which corresponds to the case of $d\,=\,0$).

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arXiv2021

Topological invariants of groups and Koszul modules

We provide a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace K inside the second exterior product of a vector space, as well as a sharp upper bound for its Hilbert function. This purely algebraic statement has interesting applications to the study of a number of invariants associated to finitely generated groups, such as the Alexander invariants, the Chen ranks, or the degree of growth and nilpotency class. For instance, we explicitly bound the aforementioned invariants in terms of the first Betti number for the maximal metabelian quotients of (1) the Torelli group associated to the moduli space of curves; (2) nilpotent fundamental groups of compact Kaehler manifolds; (3) the Torelli group of a free group.

math.GR↗
arXiv2019

Koszul modules and Green's conjecture

We prove a strong vanishing result for finite length Koszul modules, and use it to derive Green's conjecture for every g-cuspidal rational curve over an algebraically closed field k with char(k) = 0 or char(k) >= (g+2)/2. As a consequence, we deduce that the general canonical curve of genus g satisfies Green's conjecture in this range. Our results are new in positive characteristic, whereas in characteristic zero they provide a different proof for theorems first obtained in two landmark papers by Voisin. Our strategy involves establishing two key results of independent interest: (1) we describe an explicit, characteristic-independent version of Hermite reciprocity for sl_2-representations; (2) we completely characterize, in arbitrary characteristics, the (non-)vanishing behavior of the syzygies of the tangential variety to a rational normal curve.

math.AG↗
arXiv2019

On the Borisov-Nuer conjecture and the image of the Enriques-to-K3 map

We discuss the Borisov-Nuer conjecture in connection with the canonical maps from the moduli spaces $\mathcal M_{En,h}^a$of polarized Enriques surfaces with fixed polarization type $h$ to the moduli space $\mathcal F_g$ of polarized $K3$ surfaces of genus $g$ with $g=h^2+1$, and we exhibit a naturally defined locus $Σ_g\subset\mathcal F_g$. One direct consequence of the Borisov-Nuer conjecture is that $Σ_g$ would be contained in a particular Noether-Lefschetz divisor in $\mathcal F_g$, which we call the Borisov-Nuer divisor and we denote by $\mathcal{BN}_g$. In this short note, we prove that $Σ_g\cap\mathcal{BN}_g$ is non-empty whenever $(g-1)$ is divisible by $4$. To this end, we construct polarized Enriques surfaces $(Y, H_Y)$, with $H_Y^2$ divisible by $4$, which verify the conjecture. In particular, the conjecture holds also for any element $\mathcal M_{En,h}^a$, if $h^2$ is divisible by $4$ and $h$ is the same type of polarization.

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arXiv2016

Ulrich bundles on ruled surfaces

In this short note, we study the existence problem for Ulrich bundles on ruled surfaces, focusing our attention on the smallest possible rank. We show that existence of Ulrich line bundles occurs if and only if the coefficient $α$ of the minimal section in the numerical class of the polarization equals one. For other polarizations, we prove the existence of rank two Ulrich bundles.

math.AG↗
arXiv2016

Rank-two vector bundles on non-minimal ruled surfaces

We continue previous works by various authors and study the birational geometry of moduli spaces of stable rank-two vector bundles on surfaces with Kodaira dimension $-\infty$. To this end, we express vector bundles as natural extensions, by using two numerical invariants associated to vector bundles, similar to the invariants defined by Brinzanescu and Stoia in the case of minimal surfaces. We compute explicitly these natural extensions on blowups of general points on a minimal surface. In the case of rational surfaces, we prove that any irreducible component of a moduli space is either rational or stably rational.

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