Hirzebruch Surfaces Research Articles

Nef Cones of Nested Hilbert Schemes of Points on Surfaces

AbstractLet X be the projective plane, a Hirzebruch surface, or a general K3 surface. In this paper, we study the birational geometry of various nested Hilbert schemes of points parameterizing pairs of zero-dimensional subschemes on X. We calculate the nef cone for two types of nested Hilbert schemes. As an application, we recover a theorem of Butler on syzygies on Hirzebruch surfaces.

A Lie theoretical construction of a Landau–Ginzburg model without projective mirrors

We describe the Fukaya-Seidel category of a Landau-Ginzburg model LG(2) for the semisimple adjoint orbit of sl(2,C). We prove that this category is equivalent to a full triangulated subcategory of the category of coherent sheaves on the second Hirzebruch surface. We show that no projective variety can be mirror to LG(2), and that this remains so after compactification.

Enumerative geometry of elliptic curves on toric surfaces

We establish the equality of classical and tropical curve counts for elliptic curves on toric surfaces with fixed $j$-invariant, refining results of Mikhalkin and Nishinou--Siebert. As an application, we determine a formula for such counts on $\mathbb P^2$ and all Hirzebruch surfaces. This formula relates the count of elliptic curves with the number of rational curves on the surface satisfying a small number of tangency conditions with the toric boundary. Furthermore, the combinatorial tropical multiplicities of Kerber and Markwig for counts in $\mathbb P^2$ are derived and explained algebro-geometrically, using Berkovich geometry and logarithmic Gromov--Witten theory. As a consequence, a new proof of Pandharipande's formula for counts of elliptic curves in $\mathbb P^2$ with fixed $j$-invariant is obtained.

Semi-stable vector bundles on fibred varieties

We introduce and investigate a semi-stability concept relative to morphisms $$Y\xrightarrow {\scriptscriptstyle \pi } X$$ between projective manifolds, different of Maruyama–Simpson. For torsion-free sheaves on Y and for appropriate polarizations, we relate the (semi-)stability on Y to the slope (semi-)stability on the generic fibre of $$\pi $$ . We apply the result to describe in great detail the moduli spaces of semi-stable vector bundles on Hirzebruch surfaces and on threefolds which are $$\mathbb P^2$$ -bundles over . We establish their irreducibility and rationality, an intensely investigated issue.

Section rings of $\mathbb{Q}$-divisors on minimal rational surfaces

We give bounds on the degree of generation and relations of section rings associated to arbitrary $\mathbb{Q}$-divisors on projective spaces of all dimensions and Hirzebruch surfaces. For section rings of effective $\mathbb{Q}$-divisors on projective spaces, we find the best possible bound on the degrees of generators and relations.

Construction of hyperbolic Horikawa surfaces

We construct a Brody hyperbolic Horikawa surface that is a double cover of $\mathbb{P}^2$ branched along a smooth curve of degree $10$. We also construct Brody hyperbolic double covers of Hirzebruch surfaces with branch loci of the lowest possible bidegree.

On Calabi–Yau generalized complete intersections from Hirzebruch varieties and novel $K3$-fibrations

We consider the construction of Calabi-Yau varieties recently generalized to where the defining equations may have negative degrees over some projective space factors in the embedding space. Within such "generalized complete intersection" Calabi-Yau ("gCICY") three-folds, we find several sequences of distinct manifolds. These include both novel elliptic and K3-fibrations and involve Hirzebruch surfaces and their higher dimensional analogues. En route, we generalize the standard techniques of cohomology computation to these generalized complete intersection Calabi-Yau varieties.

An unbounded family of log Calabi–Yau pairs

We give an explicit example of log Calabi–Yau pairs that are log canonical and have a linearly decreasing Euler characteristic. This is constructed in terms of a degree two covering of a sequence of blow ups of three dimensional projective bundles over the Segre–Hirzebruch surfaces \mathbb F_n for every positive integer n big enough.

Corrigendum and addendum to “Moduli spaces of framed sheaves and quiver varieties”

Abstract This paper is an erratum to our paper Moduli spaces of framed sheaves and quiver varieties (J. Geom. Phys., 2016). As a byproduct, we prove a result (Prop. 2.5) providing a description of the fibre T V ∨ Gr ( a , r ) ⊕ n − 1 , for each V ∈ Gr ( a , r ) , as the space of isomorphism classes of certain extensions of sheaves on Hirzebruch surfaces.

More Toda-like (0,2) mirrors

In this paper, we extend our previous work to construct (0, 2) Toda-like mirrors to A/2-twisted theories on more general spaces, as part of a program of understanding (0,2) mirror symmetry. Specifically, we propose (0, 2) mirrors to GLSMs on toric del Pezzo surfaces and Hirzebruch surfaces with deformations of the tangent bundle. We check the results by comparing correlation functions, global symmetries, as well as geometric blowdowns with the corresponding (0, 2) Toda-like mirrors. We also briefly discuss Grassmannian manifolds.

Noncontractible Hamiltonian loops in the kernel of Seidel’s representation

The main purpose of this note is to exhibit a Hamiltonian diffeomorphism loop undetected by the Seidel morphism of certain 2-point blow-ups of $S^2 \times S^2$, exactly one of which being monotone. As side remarks, we show that Seidel's morphism is injective on all Hirzebruch surfaces and discuss how to adapt the monotone example to the Lagrangian setting.

Equigeneric and equisingular families of curves on surfaces

We investigate the following question: let $C$ be an integral curve contained in a smooth complex algebraic surface $X$; is it possible to deform $C$ in $X$ into a nodal curve while preserving its geometric genus? We affirmatively answer it in most cases when $X$ is a Del Pezzo or Hirzebruch surface, and in some cases when $X$ is a $K3$ surface. Partial results are given for all surfaces with numerically trivial canonical class. We also give various examples for which the answer is negative.

Hilbert schemes of points of [formula omitted] as quiver varieties

In a previous paper, a realization of the moduli space of framed torsion-free sheaves on Hirzebruch surfaces in terms of monads was given. We build upon that result to construct ADHM data for the Hilbert scheme of points of the total space of the line bundles O(−n) on P1, for n≥1, i.e., the resolutions of the singularities of type 1n(1,1). Basically by implementing a version of the special McKay correspondence, this ADHM description is in turn used to realize these Hilbert schemes as irreducible connected components of quiver varieties. We obtain in this way new examples of quiver varieties which are not of the Nakajima type.

Topological obstructions for rational cuspidal curves in Hirzebruch surfaces

We study rational cuspidal curves in Hirzebruch surfaces. We provide two obstructions for the existence of rational cuspidal curves in Hirzebruch surfaces with prescribed types of singular points. The first result comes from Heegaard--Floer theory and is a generalization of a result by Livingston and the first author. The second criterion is obtained by comparing the spectrum of a suitably defined link at infinity of a curve with spectra of its singular points.

Moduli spaces of framed sheaves and quiver varieties

In the first part of this paper we provide a survey of some fundamental results about moduli spaces of framed sheaves on smooth projective surfaces. In particular, we outline a result by Bruzzo and Markushevich, and discuss a few significant examples. The moduli spaces of framed sheaves on $\mathbb{P}^2$, on multiple blowup of $\mathbb{P}^2$ are described in terms of ADHM data and, when this characterization is available, as quiver varieties. The second part is devoted to a detailed study of framed sheaves on the Hirzebruch surface $\Sigma_n$ in the case when the invariant expressing the necessary and sufficient condition for the nonemptiness of moduli spaces attains its minimum (what we call the "minimal case"). Our main result is that, under this assumption, the corresponding moduli space is isomorphic to a Grassmannian (when $n=1$), or to the direct sum of $n-1$ copies of the cotangent bundle of a Grassmannian (when $n\geq 2$). Finally, by slightly generalizing a construction due to Nakajima, we prove that these moduli spaces admit a description as quiver varieties.

Hilbert schemes of some threefold scrolls over 𝔽 e

Abstract Hilbert schemes of suitable smooth, projective threefold scrolls over the Hirzebruch surface 𝔽 e , e ≥ 2, are studied. An irreducible component of the Hilbert scheme parametrizing such varieties is shown to be generically smooth of the expected dimension, and the general point of such a component is described.

The nef cones of and minimal-degree curves in the Hilbert schemes of points on certain surfaces

We determine the nef cones of the Hilbert schemes of points on certain surfaces X with h 1 .X;OX/ = 0. Then we apply the results to Hirzebruch surfaces, and study the minimal-degree curves in the Hilbert schemes of points on Hirzebruch surfaces. Our results generalize those in Li, Qin, and Zhang (2003).

ABJM on ellipsoid and topological strings

It is known that the large N expansion of the partition function in ABJM theory on a three-sphere is completely determined by the topological string on local Hirzebruch surface F_0. In this note, we investigate the ABJM partition function on an ellipsoid, which has a conventional deformation parameter b. Using 3d mirror symmetry, we find a remarkable relation between the ellipsoid partition function for b^2=3 (or b^2=1/3) in ABJM theory at k=1 and a matrix model for the topological string on another Calabi-Yau threefold, known as local P^2. As in the case of b=1, we can compute the full large N expansion of the partition function in this case. This is the first example of the complete large N solution in ABJM theory on the squashed sphere. Using the obtained results, we also analyze the supersymmetric Renyi entropy.

Linear Pencils Encoded in the Newton Polygon

Let C be an algebraic curve defined by a sufficiently generic bivariate Laurent polynomial with given Newton polygon Delta. It is classical that the geometric genus of C equals the number of lattice points in the interior of Delta. In this paper we give similar combinatorial interpretations for the gonality, the Clifford index, and the Clifford dimension, by removing a technical assumption from a recent result of Kawaguchi. More generally, the method shows that apart from certain well-understood exceptions, every base-point free pencil whose degree equals or slightly exceeds the gonality is combinatorial, in the sense that it corresponds to projecting C along a lattice direction. Along the way we prove various features of combinatorial pencils. For instance, we give an interpretation for the scrollar invariants associated with a combinatorial pencil, and show how one can tell whether the pencil is complete or not. Among the applications, we find that every smooth projective curve admits at most one Weierstrass semi-group of embedding dimension 2, and that if a non-hyperelliptic smooth projective curve C of genus g >= 2 can be embedded in the nth Hirzebruch surface H-n, then n is actually an invariant of C. This article comes along with three Magma files: basic_commands.m, gonal.m, neargonal.m

Deformation of tropical Hirzebruch surfaces and enumerative geometry

We illustrate the use of tropical methods by generalizing a formula due to Abramovich and Bertram, extended later by Vakil. Namely, we exhibit relations between enumerative invariants of the Hirzebruch surfaces Σ n \Sigma _n and Σ n + 2 \Sigma _{n+2} , obtained by deforming the first surface to the latter. Our strategy involves a tropical counterpart of deformations of Hirzebruch surfaces and tropical enumerative geometry on a tropical surface in three-space.