Projective Surface Research Articles

Negative curves of small genus on surfaces

Let X be a smooth geometrically irreducible projective surface over a field. In this paper we give an effective upper bound in terms of the Neron–Severi rank of X for the number of irreducible curves C on X with negative self-intersection and geometric genus less than $$b_1(X)/4$$, where $$b_1(X)$$ is the first etale Betti number of X. The proof involves a hyperbolic analog of the theory of spherical codes. More specifically, we relate these curves to the hyperbolic kissing number, and then prove upper and lower bounds for the hyperbolic kissing number in terms of the classical Euclidean kissing number.

A note on a smooth projective surface with Picard number 2

AbstractWe characterize the integral Zariski decomposition of a smooth projective surface with Picard number 2 to partially solve a problem of B. Harbourne, P. Pokora, and H. Tutaj‐Gasinska [Electron. Res. Announc. Math. Sci. 22 (2015), 103–108].

Semistability of Lazarsfeld–Mukai bundles via parabolic structures

Our aim in this article is to produce new examples of semistable Lazarsfeld–Mukai bundles on smooth projective surfaces X using the notion of parabolic vector bundles. In particular, we associate natural parabolic structures to any rank two (dual) Lazarsfeld–Mukai bundle and study the parabolic stability of these parabolic bundles. We also show that the orbifold bundles on Kawamata coverings of X corresponding to the above parabolic bundles are themselves certain (dual) Lazarsfeld–Mukai bundles. This gives semistable Lazarsfeld–Mukai bundles on Kawamata covers of the projective plane and of certain K3 surfaces.

A convolutional neural network method to improve efficiency and visualization in modeling driver’s visual field on roads using MLS data

This paper aims to introduce the convolutional neural network (CNN) into modeling driver’s visual field (VF) using mobile laser scanning (MLS) data. A new solution that incorporates CNN is proposed to tackle the issues of inefficiency and inadequate manners of visualization in existing methods. The method operates along vehicle trajectory recorded in MLS data. For any driver position, the initial VF is defined as a fan-shaped area originating at the driver’s viewpoint. Within the initial VF, numerous virtual line-of-sights (LOS) are emitted from the viewpoint. Given an object point in any LOS, three-dimensional (3D) MLS points that may affect its visibility are converted to two-dimensional (2D) points using the cylindrical perspective projection. 2D points on the projective surface are then transformed into a binary image via the Pixelation procedure. Fed with the generated image, the CNN which is trained based on 789,500 data will classify the visibility as: 0-visible or 1-invisible. The location of the obstacle that blocks the driver’s view along each LOS is detected with a combination of the trained CNN and the bisection method. With all positions of obstacles determined, the final VF is established. Through comparisons with a state-of-the-art method, the CNN-based method shows remarkable efficiency, which facilitates either VF modeling at a single position or successive VF analyses along the vehicle path. A case study is also presented to show the improved manners of data visualization implemented in the developed method: (1) 3D viewshed, (2) sight distance curve, and (3) the driver’s perspective image with obstacles spotlighted.

Minimal Lagrangian connections on compact surfaces

We introduce the notion of a minimal Lagrangian connection on the tangent bundle of a manifold and classify all such connections in the case where the manifold is a compact oriented surface of non-vanishing Euler characteristic. Combining our classification with results of Labourie and Loftin, we conclude that every properly convex projective surface arises from a unique minimal Lagrangian connection.

Extremal conformal structures on projective surfaces

We introduce a new functional $\mathcal{E}_{\mathfrak{p}}$ on the space of conformal structures on an oriented projective manifold $(M,\mathfrak{p})$. The nonnegative quantity $\mathcal{E}_{\mathfrak{p}}([g])$ measures how much $\mathfrak{p}$ deviates from being defined by a $[g]$-conformal connection. In the case of a projective surface $(\Sigma,\mathfrak{p})$, we canonically construct an indefinite Kahler--Einstein structure $(h_{\mathfrak{p}},\Omega_{\mathfrak{p}})$ on the total space $Y$ of a fibre bundle over $\Sigma$ and show that a conformal structure $[g]$ is a critical point for $\mathcal{E}_{\mathfrak{p}}$ if and only if a certain lift $\widetilde{[g]} : (\Sigma,[g]) \to (Y,h_{\mathfrak{p}})$ is weakly conformal. In fact, in the compact case $\mathcal{E}_{\mathfrak{p}}([g])$ is -- up to a topological constant -- just the Dirichlet energy of $\widetilde{[g]}$. As an application, we prove a novel characterisation of properly convex projective structures among all flat projective structures. As a by-product, we obtain a Gauss--Bonnet type identity for oriented projective surfaces.

Counting lines on surfaces, especially quintics

We introduce certain rational functions on a smooth projective surface X in IP^3 which facilitate counting the lines on X. We apply this to smooth quintics in characteristic zero to prove that they contain no more than 127 lines, and that any given line meets at most 28 others. We construct examples which demonstrate that the latter bound is sharp.

Automorphisms of Hilbert schemes of points on a generic projective K3 surface

AbstractWe study automorphisms of the Hilbert scheme of n points on a generic projective K3 surface S, for any . We show that is either trivial or generated by a non‐symplectic involution and we determine numerical and divisorial conditions which allow us to distinguish between the two cases. As an application of these results we prove that, for any , there exist infinitely many admissible degrees for the polarization of the K3 surface S such that admits a non‐natural involution. This provides a generalization of the results of [7] for .

Commensurability classes of fake quadrics

A fake quadric is a smooth projective surface that has the same rational cohomology as a smooth quadric surface but is not biholomorphic to one. We provide an explicit classification of all irreducible fake quadrics according to the commensurability class of their fundamental group. To accomplish this task, we develop a number of new techniques that explicitly bound the arithmetic invariants of a fake quadric and more generally of an arithmetic manifold of bounded volume arising from a form of $${{\,\mathrm{SL}\,}}_2$$ over a number field.

Effective algebraic integration in bounded genus

We introduce and study birational invariants for foliations on projective surfaces built from the adjoint linear series of positive powers of the canonical bundle of the foliation. We apply the results in order to investigate the effective algebraic integration of foliations on the projective plane. In particular, we describe the Zariski closure of the set of foliations on the projective plane of degree d admitting rational first integrals with fibers having geometric genus bounded by g.

The degree of the tangent and secant variety to a projective surface

Abstract We present a way of computing the degree of the secant (resp. tangent) variety of a smooth projective surface, under the assumption that the divisor giving the embedding in the projective space is 3-very ample. This method exploits the link between these varieties and the Hilbert scheme 0-dimensional subschemes of length 2 of the surface.

ON VARIETIES WITH TRIVIAL TANGENT BUNDLE IN CHARACTERISTIC

In this article, I give a crystalline characterization of abelian varieties amongst the class of smooth projective varieties with trivial tangent bundles in characteristic $p>0$. Using my characterization, I show that a smooth, projective, ordinary variety with trivial tangent bundle is an abelian variety if and only if its second crystalline cohomology is torsion-free. I also show that a conjecture of KeZheng Li about smooth projective varieties with trivial tangent bundles in characteristic $p>0$ is true for smooth projective surfaces. I give a new proof of a result by Li and prove a refinement of it. Based on my characterization of abelian varieties, I propose modifications of Li’s conjecture, which I expect to be true.

On the irreducibility of the Severi variety of nodal curves in a smooth surface

Let X be a smooth projective surface and $$L\in \mathrm {Pic}(X)$$ . We prove that if L is $$(2k-1)$$ -spanned, then the set $${\tilde{V}}_k(L)$$ of all nodal and irreducible $$D\in |L|$$ with exactly k nodes is irreducible. The set $${\tilde{V}}_k(L)$$ is an open subset of a Severi variety of |L|, the full Severi variety parametrizing all integral $$D\in |L|$$ with geometric genus $$g(L)-k$$ .

Shuffle algebras associated to surfaces

We consider the algebra of Hecke correspondences (elementary transformations at a single point) acting on the algebraic K-theory groups of the moduli spaces of stable sheaves on a smooth projective surface S. We derive quadratic relations between the Hecke correspondences, and compare the algebra they generate with the Ding–Iohara–Miki algebra (at a suitable specialization of parameters), as well as with a generalized shuffle algebra.

K3 surfaces with Picard number 2, Salem polynomials and Pell equation

If an automorphism of a projective K3 surface with Picard number 2 is of infinite order, then the automorphism corresponds to a solution of Pell equation. In this paper, by solving this equation, we determine all Salem polynomials of symplectic and anti-symplectic automorphisms of projective K3 surfaces with Picard number 2.

Formality conjecture for K3 surfaces

We give a proof of the formality conjecture of Kaledin and Lehn: on a complex projective K3 surface, the differential graded (DG) algebra$\operatorname{RHom}^{\bullet }(F,F)$is formal for any sheaf$F$polystable with respect to an ample line bundle. Our main tool is the uniqueness of the DG enhancement of the bounded derived category of coherent sheaves. We also extend the formality result to derived objects that are polystable with respect to a generic Bridgeland stability condition.

Affine surfaces with isomorphic A2-cylinders

We show that all complements of cuspidal hyperplane sections of smooth projective cubic surfaces have isomorphic A2-cylinders. As a consequence, we derive that the A2-cancellation problem fails in every dimension greater than or equal to 2.

Representations of braid groups and construction of projective surfaces

Braid groups are an important and flexible tool used in several areas of science, such as Knot Theory (Alexander’s theorem), Mathematical Physics (Yang-Baxter’s equation) and Algebraic Geometry (monodromy invariants). In this note we will focus on their algebraic-geometric aspects, explaining how the representation theory of higher genus braid groups can be used to produce interesting examples of projective surfaces defined over the field of complex numbers.

Some Singular Curves and Surfaces Arising from Invariants of Complex Reflection Groups

We construct highly singular projective curves and surfaces defined by invariants of primitive complex reflection groups. As a result, we improve some lower bounds on the number of singularities of a given type that a plane curve or a surface in of a given degree might have.

Resolution of singularities of arithmetical threefolds

We prove Grothendieck's conjecture on Resolution of Singularities for quasi-excellent schemes X of dimension three and of arbitrary characteristic. This applies in particular to X=SpecA, A a reduced complete Noetherian local ring of dimension three and to algebraic or arithmetical varieties of dimension three. Similarly, if F is a number field, a complete discretely valued field or more generally the quotient field of any excellent Dedekind domain O, any regular projective surface X/F has a proper and flat model X over O which is everywhere regular.