Searcharxiv⌕ Search

arXiv subjects

Howard Nuer

Publications and source records attributed to Howard Nuer.

16 recordsLinked to original sources
arXiv2026

The equations of general Hassett maximal cubic fourfolds

In this note, we discuss Hassett maximal cubic fourfolds and construct an explicit irreducible component of maximal dimension sixteen of the locus $\mathcal{Z}$ of Hassett maximal cubic fourfolds. We utilize algebraic and arithmetic methods to analyze the associated lattice of these fourfolds. % By studying general integral quadratic forms and proving the ADC property for a specific ternary form, we demonstrate that the primitive image of our lattice spans the entire Hassett subset, confirming the Hassett maximality of the cubic fourfolds we describe.

math.AG↗
arXiv2026

Finite order symplectic birational self-maps on Kummer-type manifolds

A projective hyperk\"ahler manifold of Kummer-type is said to be twisted modular if it is birational to the Albanese fiber of a moduli space of twisted sheaves on an abelian surface. We prove that, with the exception of certain cases of Picard rank 3, any projective Kummer-type manifold admitting a finite-order symplectic birational self-map that acts nontrivially on its second cohomology group is twisted modular. We provide a complete characterization of these exceptions in terms of their N\'eron-Severi lattices. We then investigate symplectic birational self-maps of modular Kummer-type manifolds, determining exactly which Mukai vectors allow the birational transformation induced by crossing the vertical wall, which acts on cohomology as a reflection, to correspond to a finite-order symplectic birational self-map. Additionally, we prove in an appendix several results concerning moduli spaces of twisted sheaves on abelian surfaces which were not readily available in the literature.

math.AG↗
arXiv2024

Weak Brill-Noether on Abelian Surfaces

We study the cohomology of a general stable sheaf on an abelian surface. We say that a moduli space satisfies weak Brill-Noether if the general sheaf has at most one non-zero cohomology group. Let $(X,H)$ be a polarized abelian surface and let $\mathbf{v}=(r,\xi,a)$ be a Mukai vector on $X$ with $\mathbf{v}^2\ge 0$,$r>0$, and $\xi\cdot H>0$. We show that if $\rho(X)=1$ or $\rho(X)=2$ and $X$ contains an elliptic curve, then all the moduli spaces $M_{X,H}(\mathbf{v})$ satisfy weak Brill-Noether. Conversely, if $\rho(X)>2$ or $\rho(X)=2$ and $X$ does not contain an elliptic curve, we show that there are infinitely many moduli spaces $M_{X,H}(\mathbf{v})$ that fail weak Brill-Noether. As a consequence, we classify Chern classes of Ulrich bundles on abelian surfaces.

math.AG↗
arXiv2023

Supporting rank and the intersection of all Hassett Divisors

We prove that the dimension of the intersection $\mathcal Z$ of all Hassett divisors of special cubic fourfolds is sixteen. We do this by studying which subsets of the natural numbers $\mathbb N$ can be obtained as the image of a positive-definite integral quadratic form and what the minimal possible rank of such a form is. In particular, for the subset of $\mathbb N$ consisting of all possible discriminants of special cubic fourfolds, we show this rank is four and that this is the codimension of $\mathcal Z$ in $\mathcal C$, the twenty-dimensional moduli space of cubic fourfolds.

math.AG↗
arXiv2022

Stability conditions in families

We develop a theory of Bridgeland stability conditions and moduli spaces of semistable objects for a family of varieties. Our approach is based on and generalizes previous work by Abramovich-Polishchuk, Kuznetsov, Lieblich, and Piyaratne-Toda. Our notion includes openness of stability, semistable reduction, a support property uniformly across the family, and boundedness of semistable objects. We show that such a structure exists whenever stability conditions are known to exist on the fibers. Our main application is the generalization of Mukai's theory for moduli spaces of semistable sheaves on K3 surfaces to moduli spaces of Bridgeland semistable objects in the Kuznetsov component associated to a cubic fourfold. This leads to the extension of theorems by Addington-Thomas and Huybrechts on the derived category of special cubic fourfolds, to a new proof of the integral Hodge conjecture, and to the construction of an infinite series of unirational locally complete families of polarized hyperkähler manifolds of K3 type. Other applications include the deformation-invariance of Donaldson-Thomas invariants counting Bridgeland stable objects on Calabi-Yau threefolds, and a method for constructing stability conditions on threefolds via degeneration.

math.AG↗
arXiv2021

Stable sheaves on bielliptic surfaces: from the classical to the modern

Bielliptic surfaces are the last family of Kodaira dimension zero algebraic surfaces without a classification result for the Chern characters of stable sheaves. We rectify this and prove such a classification using a combination of classical techniques, on the one hand, and derived category and Bridgeland stability techniques, on the other. Along the way, we prove the existence of projective coarse moduli spaces of objects in the derived category of a bielliptic surface that Bridgeland semistable with respect to a generic stability condition. By systematically studying the connection between Bridgeland wall-crossing and birational geometry, we show that for any two generic stability conditions $τ,σ$, the two moduli spaces $M_τ(\mathbf{v})$ and $M_σ(\mathbf{v})$ of objects of Chern character $\mathbf{v}$ that are semistable with respect to $τ$ (resp. $σ$) are birational. As a consequence, we show that for primitive $\mathbf{v}$, the moduli space of stable sheaves of class $\mathbf{v}$ is birational to a moduli space of stable sheaves whose Chern character has one of finitely many easily understood "shapes".

math.AG↗
arXiv2021

The cohomology of the general stable sheaf on a K3 surface

Let $X$ be a K3 surface with Picard group $\mathrm{Pic}(X)\cong\mathbb{Z} H$ such that $H^2=2n$. Let $M_{H}(\mathbf{v})$ be the moduli space of Gieseker semistable sheaves on $X$ with Mukai vector $\mathbf{v}$. We say that $\mathbf{v}$ satisfies weak Brill-Noether if the general sheaf in $M_{H}(\mathbf{v})$ has at most one nonzero cohomology group. We show that given any rank $r \geq 2$, there are only finitely many Mukai vectors of rank $r$ on K3 surfaces of Picard rank one where weak Brill-Noether fails. We give an algorithm for finding the potential counterexamples and classify all such counterexamples up to rank 20 explicitly. Moreover, in each of these cases we calculate the cohomology of the general sheaf. Given $r$, we give sharp bounds on $n$, $d$, and $a$ that guarantee that $\mathbf{v}=(r,dH,a)$ satisfies weak Brill-Noether. As a corollary, we obtain another proof of the classification of Ulrich bundles on K3 surfaces of Picard rank one. In addition, we discuss the question of when the general sheaf in $M_H(\mathbf{v})$ is globally generated. Our techniques make crucial use of Bridgeland stability conditions.

math.AG↗
arXiv2020

A refined Derived Torelli Theorem for Enriques surfaces

We prove that two general Enriques surfaces defined over an algebraically closed field of characteristic different from $2$ are isomorphic if their Kuznetsov components are equivalent. We apply the same techniques to give a new simple proof of a conjecture by Ingalls and Kuznetsov relating the derived categories of the blow-up of general Artin\textendash Mumford quartic double solids and of the associated Enriques surfaces. This paper originated from one of the problem sections at the workshop \emph{Semiorthogonal decompositions, stability conditions and sheaves of categories}, Université de Toulouse, May 2--5, 2018.

math.AG↗
arXiv2019

MMP Via Wall-crossing for Moduli Spaces of Stable Sheaves on an Enriques surface

We use wall-crossing in the Bridgeland stability manifold to systematically study the birational geometry of the moduli space $M_σ(\mathbf{v})$ of $σ$-semistable objects of class $\mathbf{v}$ for a generic stability condition $σ$ on an arbitrary Enriques surface $X$. In particular, we show that for any other generic stability condition $τ$, the two moduli spaces $M_τ(\mathbf{v})$ and $M_σ(\mathbf{v})$ are birational. As a consequence, we show that for primitive $\mathbf{v}$ of odd rank $M_σ(\mathbf{v})$ is birational to a Hilbert scheme of points. Similarly, in even rank we show that $M_σ(\mathbf{v})$ is birational to a moduli space of torsion sheaves supported on a hyperelliptic curve when $\ell(\mathbf{v})=1$. As an added bonus of our work, we prove that the Donaldson-Mukai map $θ_{\mathbf{v},σ}:\mathbf{v}^\perp\to\mathrm{Pic}(M_σ(\mathbf{v}))$ is an isomorphism for these classes. Finally, we use our classification to fully describe the geometry of the only two examples of moduli of stable sheaves on $X$ that are uniruled (and thus not K-trivial).

math.AG↗
arXiv2016

Unirationality of moduli spaces of special cubic fourfolds and K3 surfaces

We provide explicit descriptions of the generic members of Hassett's divisors $\mathcal C_d$ for relevant $18\leq d\leq 38$ and for $d=44$. In doing so, we prove that $\mathcal C_d$ is unirational for $18\leq d\leq 38,d=44$. As a corollary, we prove that the moduli space $\mathcal N_{d}$ of polarized K3 surfaces of degree $d$ is unirational for $d=14,26,38$. The case $d=26$ is entirely new, while the other two cases have been previously proven by Mukai.

math.AG↗
arXiv2016

Ulrich Bundles on Enriques Surfaces

We study Ulrich bundles and their moduli on unnodal Enriques surfaces. In particular, we prove that unnodal Enriques surfaces are of wild representation type by constructing moduli spaces of stable Ulrich bundles of arbitrary rank and arbitrarily large dimension.

math.AG↗
arXiv2014

Projectivity and Birational Geometry of Bridgeland Moduli spaces on an Enriques Surface

We construct moduli spaces of semistable objects on an Enriques surface for generic Bridgeland stability condition and prove their projectivity. We further generalize classical results about moduli spaces of semistable sheaves on an Enriques surface to their Bridgeland counterparts. Using Bayer and Macrì's construction of a natural nef divisor varying with the stability condition, we begin a systematic exploration of the relation between wall-crossing on the Bridgeland stability manifold and the minimal model program for these moduli spaces. We give three applications of our machinery to obtain new information about the classical moduli spaces of Gieseker-stable sheaves: 1) We obtain a region in the ample cone of the moduli space of Gieseker-stable sheaves which works for all unnodal Enriques surfaces. 2) We determine the nef cone of the Hilbert scheme of $n$ points on an unnodal Enriques surface in terms of the classical geometry of its half-pencils and the Cossec-Dolgachev $ϕ$-function. 3) We recover some classical results on linear systems on Enriques surfaces and obtain some new ones about $n$-very ample line bundles.

math.AG↗
arXiv2014

A Note on the existence of stable vector bundles on Enriques surfaces

We prove the non-emptiness of $M_{H,Y}(v)$, the moduli space of Gieseker-semistable sheaves on an unnodal Enriques surface $Y$ with Mukai vector $v$ of positive rank with respect to a generic polarization $H$. This completes the chain of progress initiated by H. Kim in \cite{Kim98}. We also show that the stable locus $M^s_{H,Y}(v)\neq\varnothing$ for $v^2>0$. Finally, we prove irreducibility of $M_{H,Y}(v)$ in case $v^2=0$ and $v$ primitive.

math.AG↗