Projective Surface Research Articles

On the standard conjecture for a -dimensional variety fibred by curves with a non-injective Kodaira–Spencer map

We prove that the Grothendieck standard conjecture of Lefschetz type holds for a complex projective 3-dimensional variety fibred by curves (possibly with degeneracies) over a smooth projective surface provided that the endomorphism ring of the Jacobian variety of some smooth fibre coincides with the ring of integers and the corresponding Kodaira–Spencer map has rank on some non-empty open subset of the surface. When the generic fibre of the structure morphism is of genus , the condition on the endomorphisms of the Jacobian may be omitted.

Fujita-type freeness for quasilog canonical curves and surfaces

We prove Fujita-type basepoint-freeness for projective quasi-log canonical curves and surfaces.

On the formal principle for curves on projective surfaces

We prove that the formal completion of a complex projective surface along a rigid smooth curve with trivial normal bundle determines the birational equivalence class of the surface.

A surface in odd characteristic with discrete and non-finitely generated automorphism group

It was proved by Tien-Cuong Dinh and me that there is a smooth complex projective surface whose automorphism group is discrete and not finitely generated. In this paper, after observing finite generation of the automorphism group of any smooth projective surface birational to any K3 surface over any algebraic closure of the prime field of odd characteristic, we will show that there is a smooth projective surface, birational to some K3 surface, such that the automorphism group is discrete and not finitely generated, over any algebraically closed field of odd characteristic of positive transcendental degree over the prime field.

Cox rings of K3 surfaces of Picard number three

Let X be a projective K3 surface over C. We prove that its Cox ring has a generating set whose degrees are either classes of smooth rational curves, sums of at most three elements of the Hilbert basis of the nef cone, or of the form 2(f+f′), where f,f′ are classes of smooth elliptic curves with f⋅f′=2. This result and techniques using Koszul's type exact sequences are then applied to determine a generating set for the Cox ring of all Mori dream K3 surfaces of Picard number three which is minimal in most cases. A presentation for the Cox ring is given in some special cases with few generators.

Metrisability of projective surfaces and pseudo-holomorphic curves

We show that the metrisability of an oriented projective surface is equivalent to the existence of pseudo-holomorphic curves. A projective structure {mathfrak {p}} and a volume form sigma on an oriented surface M equip the total space of a certain disk bundle Zrightarrow M with a pair (J_{{mathfrak {p}}},{mathfrak {J}}_{{mathfrak {p}},sigma }) of almost complex structures. A conformal structure on M corresponds to a section of Zrightarrow M and {mathfrak {p}} is metrisable by the metric g if and only if [g] : M rightarrow Z is a pseudo-holomorphic curve with respect to J_{{mathfrak {p}}} and {mathfrak {J}}_{{mathfrak {p}},dA_g}.

Fundamental group schemes of Hilbert scheme of n points on a smooth projective surface

Let k be an algebraically closed field of characteristic p>3. Let X be an irreducible smooth projective surface over k. Fix an integer n≥2 and let HilbXn be the Hilbert scheme parameterizing effective 0-cycles of length n on X. The aim of the present article is to find the S-fundamental group scheme and Nori's fundamental group scheme of the Hilbert scheme HilbXn.

Degeneracy loci, virtual cycles and nested Hilbert schemes II

We express nested Hilbert schemes of points and curves on a smooth projective surface as ‘virtual resolutions’ of degeneracy loci of maps of vector bundles on smooth ambient spaces. We show how to modify the resulting obstruction theories to produce the virtual cycles of Vafa–Witten theory and other sheaf-counting problems. The result is an effective way of calculating invariants (VW, SW, local PT and local DT) via Thom–Porteous-like Chern class formulae.

LEHN’S FORMULA IN CHOW AND CONJECTURES OF BEAUVILLE AND VOISIN

AbstractThe Beauville–Voisin conjecture for a hyperkähler manifold $X$ states that the subring of the Chow ring $A^{\ast }(X)$ generated by divisor classes and Chern characters of the tangent bundle injects into the cohomology ring of $X$. We prove a weak version of this conjecture when $X$ is the Hilbert scheme of points on a K3 surface for the subring generated by divisor classes and tautological classes. This in particular implies the weak splitting conjecture of Beauville for these geometries. In the process, we extend Lehn’s formula and the Li–Qin–Wang $W_{1+\infty }$ algebra action from cohomology to Chow groups for the Hilbert scheme of an arbitrary smooth projective surface $S$.

Equivariant K-Theory and Refined Vafa\u2013Witten Invariants

In Maulik and Thomas (in preparation) the Vafa–Witten theory of complex projective surfaces is lifted to oriented {mathbb {C}}^*-equivariant cohomology theories. Here we study the K-theoretic refinement. It gives rational functions in t^{1/2} invariant under t^{1/2}leftrightarrow t^{-1/2} which specialise to numerical Vafa–Witten invariants at t=1. On the “instanton branch” the invariants give the virtual chi ^{}_{-t}-genus refinement of Göttsche–Kool, extended to allow for strictly semistable sheaves. Applying modularity to their calculations gives predictions for the contribution of the “monopole branch”. We calculate some cases and find perfect agreement. We also do calculations on K3 surfaces, finding Jacobi forms refining the usual modular forms, proving a conjecture of Göttsche–Kool. We determine the K-theoretic virtual classes of degeneracy loci using Eagon–Northcott complexes, and show they calculate refined Vafa–Witten invariants. Using this Laarakker (Monopole contributions to refined Vafa–Witten invariants. arXiv:1810.00385) proves universality results for the invariants.

On theK-Theoretic Hall Algebra of a Surface

AbstractIn this paper, we define the $K$-theoretic Hall algebra for dimension $0$ coherent sheaves on a smooth projective surface, prove that the algebra is associative, and construct a homomorphism to a shuffle algebra introduced by Negut [ 10].

A FOTOTERÁPIÁS ESZKÖZÖK KREATÍV ALKALMAZÁSA AZ OKTATÁSBAN

The Educational Service functions in interdisciplinary teams. The services are primarily used by children with special educational needs, along with their parents and educators. The best practice was made by thinking in an integrated service model, based on solution-focused, children – and family-oriented experiences. One of the cornerstones is coaching, especially those tools, that inspire phototherapeutic self-knowledge and the solutions of problems, Points of You™ (Efrat Shani & Yaron Golan, 2007). The phototherapeutic picture – and word cards function as projective surfaces. They stimulate both of the cerebral hemispheres at the same time, causing an intentional confusion between the logical left hemisphere and the intuitive right one. That’s when the person is able to see a new point of view and have an „aha experience”. The new point of view can already lead to the recognition of new possibilities. The Educational Service of Hajdú-Bihar County introduced the phototherapeutic method to its educational practices in a lot of areas of the service, innovatively and uniquely in the country, on an organizational level. In the area of educational and psychological counseling and development, this method can be used in individual, couple, and group forms, as well. People can use it to develop themselves, it can be used with children, parents, educators, and other processes. The core institution of the Ministry of Education has been making this best practice popular for years now.

Automorphisms of Hilbert schemes of points on surfaces

We show that every automorphism of the Hilbert scheme of n n points on a weak Fano or general type surface is natural, i.e., induced by an automorphism of the surface, unless the surface is a product of curves and n = 2 n=2 . In the exceptional case there exists a unique nonnatural automorphism. More generally, we prove that any isomorphism between Hilbert schemes of points on smooth projective surfaces, where one of the surfaces is weak Fano or of general type and not equal to the product of curves, is natural. We also show that every automorphism of the Hilbert scheme of 2 2 points on P n \mathbb {P}^n is natural.

Stability Conditions for Restrictions of Vector Bundles on Projective Surfaces

Using Bridgeland stability conditions, we give sufficient criteria for a stable vector bundle on a smooth complex projective surface to remain stable when restricted to a curve. We give a stronger criterion when the vector bundle is a general vector bundle on the plane. As an application, we compute the cohomology of such bundles for curves that lie in the plane or on Hirzebruch surfaces.

Simple groups of birational transformations in dimension two

We classify simple groups that act by birational transformations on compact complex Kähler surfaces. Moreover, we show that every finitely generated simple group that acts nontrivially by birational transformations on a projective surface over an arbitrary field is finite.

Convex projective surfaces with compatible Weyl connection are hyperbolic

We show that a properly convex projective structure $\mathfrak{p}$ on a closed oriented surface of negative Euler characteristic arises from a Weyl connection if and only if $\mathfrak{p}$ is hyperbolic. We phrase the problem as a non-linear PDE for a Beltrami differential by using that $\mathfrak{p}$ admits a compatible Weyl connection if and only if a certain holomorphic curve exists. Turning this non-linear PDE into a transport equation, we obtain our result by applying methods from geometric inverse problems. In particular, we use an extension of a remarkable $L^2$-energy identity known as Pestov's identity to prove a vanishing theorem for the relevant transport equation.

Characteristic Points, Fundamental Cubic Form and Euler Characteristic of Projective Surfaces

We define local indices for projective umbilics and godrons (also called cusps of Gauss) on generic smooth surfaces in projective 3-space. By means of these indices, we provide formulas that relate the algebraic numbers of those characteristic points on a surface (and on domains of the surface) with the Euler characteristic of that surface (resp. of those domains). These relations determine the possible coexis-tences of projective umbilics and godrons on the surface. Our study is based on a fundamental cubic form for which we provide a simple closed expression.

Some noncommutative minimal surfaces

In the ongoing programme to classify noncommutative projective surfaces (connected graded noetherian domains of Gelfand-Kirillov dimension three) a natural question is to determine the minimal models within any birational class.In this paper we show that the generic noncommutative projective plane (corresponding to the three dimensional Sklyanin algebra) as well as noncommutative analogues of P1×P1 and the more general Van den Bergh quadrics satisfy very strong minimality conditions. Translated into an algebraic question, where one is interested in a maximality condition, we prove the following result.Theorem A:Let R be a Sklyanin algebra or a Van den Bergh quadric that is infinite dimensional over its centre and letA⊇Rbe any connected graded noetherian maximal order, with the same graded quotient ring as R. Then, up to taking Veronese rings, A is isomorphic to R.Let T be an elliptic algebra (that is, the coordinate ring of a noncommutative surface containing an elliptic curve). Then, under an appropriate homological condition, we prove that every connected graded noetherian overring of T is obtained by blowing down finitely many lines (line modules) of self-intersection(−1).

Weyl and Zariski chambers on projective surfaces

Abstract Let X be a nonsingular complex projective surface. The Weyl and Zariski chambers give two interesting decompositions of the big cone of X. Following the ideas of [T. Bauer and M. Funke, Weyl and Zariski chambers on K3 surfaces, Forum Math. 24 2012, 3, 609–625] and [S. A. Rams and T. Szemberg, When are Zariski chambers numerically determined?, Forum Math. 28 2016, 6, 1159–1166], we study these two decompositions and determine when a Weyl chamber is contained in the interior of a Zariski chamber and vice versa. We also determine when a Weyl chamber can intersect non-trivially with a Zariski chamber.

Atiyah class and sheaf counting on local Calabi Yau fourfolds

We discuss Donaldson-Thomas (DT) invariants of torsion sheaves with 2 dimensional support on a smooth projective surface in an ambient non-compact Calabi Yau fourfold given by the total space of a rank 2 bundle on the surface. We prove that in certain cases, when the rank 2 bundle is chosen appropriately, the universal truncated Atiyah class of these codimension 2 sheaves reduces to one, defined over the moduli space of such sheaves realized as torsion codimension 1 sheaves in a noncompact divisor (threefold) embedded in the ambient fourfold. Such reduction property of universal Atiyah class enables us to relate our fourfold DT theory to a reduced DT theory of a threefold and subsequently then to the moduli spaces of sheaves on the base surface using results in [15,16]. We finally make predictions about modularity of such fourfold invariants when the base surface is an elliptic K3.