Minimal Surface Of General Type Research Articles

On the Hessian of Cubic Hypersurfaces

Abstract In this paper, we analyze the Hessian locus associated to a general cubic hypersurface, by describing its singular locus and its desingularization for every dimension. The strategy is based on strong connections between the Hessian and the quadrics defined as partial derivatives of the cubic polynomial. In particular, we focus our attention on the singularities of the Hessian hypersurface associated to the general cubic four-fold. It turns out to be a minimal surface of general type: its analysis is developed by exploiting the nature of this surface as a degeneracy locus of a symmetric vector bundle map and by describing an unramified double cover, which is constructed in a more general setting.

Unimodal singularities and boundary divisors in the KSBA moduli of a class of Horikawa surfaces

AbstractSmooth minimal surfaces of general type with , , and constitute a fundamental example in the geography of algebraic surfaces, and the 28‐dimensional moduli space of their canonical models admits a modular compactification via the minimal model program. We describe eight new irreducible boundary divisors in such compactification parameterizing reducible stable surfaces. Additionally, we study the relation with the GIT compactification of and the Hodge theory of the degenerate surfaces that the eight divisors parameterize.

Some surfaces with canonical map of degree 4

In this note, we construct unbounded families of minimal surfaces of general type whose canonical map has a degree of 4 such that the limits of the slopes K^2/\chi assume countably many different values in the closed interval [6+\frac23 ,8] .

Geography of minimal surfaces of general type with Z22$\mathbb {Z}_2^2$‐actions and the locus of Gorenstein stable surfaces

AbstractIn this note, the geography of minimal surfaces of general type admitting ‐actions is studied. More precisely, it is shown that Gieseker's moduli space contains surfaces admitting a ‐action for every admissible pair such that or . The examples considered allow to prove that the locus of Gorenstein stable surfaces is not closed in the KSBA‐compactification of Gieseker's moduli space for every admissible pair such that .

Simple fibrations in -surfaces

Abstract We introduce the notion of a simple fibration in $(1,2)$ -surfaces – that is, a hypersurface inside a certain weighted projective space bundle over a curve such that the general fibre is a minimal surface of general type with $p_g=2$ and $K^2=1$ . We prove that almost all Gorenstein simple fibrations over the projective line with at worst canonical singularities are canonical threefolds ‘on the Noether line’ with $K^3=\frac 43 p_g-\frac {10}3$ , and we classify them. Among them, we find all the canonical threefolds on the Noether line that have previously appeared in the literature. The Gorenstein simple fibrations over ${\mathbb {P}}^1$ are Cartier divisors in a toric $4$ -fold. This allows to us to show, among other things, that the previously known canonical threefolds on the Noether line form an open subset of the moduli space of canonical threefolds, that the general element of this component is a Mori Dream Space and that there is a second component when the geometric genus is congruent to $6$ modulo $8$ ; the threefolds in this component are new.

Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points

We study tautological vector bundles over the Hilbert scheme of points on surfaces. For each K-trivial surface, we write down a simple criterion ensuring that the tautological bundles are big and nef, and illustrate it by examples. In the K3 case, we extend recent constructions and results of Bini, Boissi\`ere and Flamini from the Hilbert scheme of 2 and 3 points to an arbitrary number of points. Among the K-trivial surfaces, the case of Enriques surfaces is the most involved. Our techniques apply to other smooth projective surfaces, including blowups of K3s and minimal surfaces of general type, as well as to the punctual Quot schemes of curves.

Extra Connected Components in the Moduli Space of Stable Surfaces

Abstract We construct examples of non-smoothable stable surface with invariants in a wide range comprising all possible invariants of smooth minimal surfaces of general type but non-standard second plurigenus. We deduce that the moduli space of stable surfaces $\overline {{\mathfrak {M}}}_{k,l}$ has at least $k-l+2$ connected components distinguished by the second plurigenus and, in particular, always has more irreducible components than the Gieseker moduli space with the same invariants.

Quot schemes of curves and surfaces: virtual classes, integrals, Euler characteristics

We compute tautological integrals over Quot schemes on curves and surfaces. After obtaining several explicit formulas over Quot schemes of dimension 0 quotients on curves (and finding a new symmetry), we apply the results to tautological integrals against the virtual fundamental classes of Quot schemes of dimension 0 and 1 quotients on surfaces (using also universality, torus localization, and cosection localization). The virtual Euler characteristics of Quot schemes of surfaces, a new theory parallel to the Vafa-Witten Euler characteristics of the moduli of bundles, is defined and studied. Complete formulas for the virtual Euler characteristics are found in the case of dimension 0 quotients on surfaces. Dimension 1 quotients are studied on K3 surfaces and surfaces of general type with connections to the Kawai-Yoshioka formula and the Seiberg-Witten invariants respectively. The dimension 1 theory is completely solved for minimal surfaces of general type admitting a nonsingular canonical curve. Along the way, we find a new connection between weighted tree counting and multivariate Fuss-Catalan numbers which is of independent interest.

Some examples of algebraic surfaces with canonical map of degree 20

In this note, we construct two minimal surfaces of general type with geometric genus p_g= 3, irregularity q = 0, self-intersection of the canonical divisor K^22 =20,24 such that their canonical map is of degree 20. In one of these surfaces, the canonical linear system has a non-trivial fixed part. These surfaces, to our knowledge, are the first examples of minimal surfaces of general type with canonical map of degree 20.

Virtual χ−y‐genera of Quot schemes on surfaces

This paper studies the virtual $\chi_{-y}$-genera of Grothendieck's Quot schemes on surfaces, thus refining the calculations of the virtual Euler characteristics by Oprea-Pandharipande. We first prove a structural result expressing the equivariant virtual $\chi_{-y}$-genera of Quot schemes universally in terms of the Seiberg-Witten invariants. The formula is simpler for curve classes of Seiberg-Witten length $N$, which are defined in the paper. By way of application, we give complete answers in the following cases: (i) arbitrary surfaces for the zero curve class, (ii) relatively minimal elliptic surfaces for rational multiples of the fiber class, (iii) minimal surfaces of general type with $p_g>0$ for any curve classes. Furthermore, a blow up formula is obtained for curve classes of Seiberg-Witten length $N$. As a result of these calculations, we prove that the generating series of the virtual $\chi_{-y}$-genera are given by rational functions for all surfaces with $p_g>0$, addressing a conjecture of Oprea-Pandharipande. In addition, we study the reduced $\chi_{-y}$-genera for $K3$ surfaces and primitive curve classes with connections to the Kawai-Yoshioka formula.

Crystalline aspects of geography of low dimensional varieties I: numerology

This is a modest attempt to study, in a systematic manner, the structure of low dimensional varieties in positive characteristics using $p$-adic invariants. The main objects of interest in this paper are surfaces and threefolds. It is known that there are many (counter) examples of `pathological' or unexpected behavior in surfaces and even of threefolds. Our focus has been on, obtaining systematically, general positive results instead of counter examples. There are many results we prove in this paper and not all can be listed in this abstract. Here are some of the results we prove: We show that $c_1^2\leq 5c_2+6b_1$ holds for a large class of surfaces (of general type). We prove that for a smooth, projective, Hodge-Witt, minimal surface of general type (with additional assumptions such as slopes of Frobenius on $H^2_{cris}(X)$ are $\geq\frac{1}{2}$) that $$c_1^2\leq 5c_2.$$ Novelty of our method lies in our use of slopes of Frobenius (and Hodge-Witt assumption which allows us to control the slope spectral sequence). We also construct new birational invariants of surfaces. Applying our methods to threefolds, we characterise Calabi-Yau threefolds with $b_3=0$.

Virtual Refinements of the Vafa\u2013Witten Formula

We conjecture a formula for the generating function of virtual chi _y-genera of moduli spaces of rank 2 sheaves on arbitrary surfaces with holomorphic 2-form. Specializing the conjecture to minimal surfaces of general type and to virtual Euler characteristics, we recover (part of) a formula of C. Vafa and E. Witten. These virtual chi _y-genera can be written in terms of descendent Donaldson invariants. Using T. Mochizuki’s formula, the latter can be expressed in terms of Seiberg–Witten invariants and certain explicit integrals over Hilbert schemes of points. These integrals are governed by seven universal functions, which are determined by their values on {mathbb {P}}^2 and {mathbb {P}}^1 times {mathbb {P}}^1. Using localization we calculate these functions up to some order, which allows us to check our conjecture in many cases. In an appendix by H. Nakajima and the first named author, the virtual Euler characteristic specialization of our conjecture is extended to include mu -classes, thereby interpolating between Vafa–Witten’s formula and Witten’s conjecture for Donaldson invariants.

Bounds on Wahl singularities from symplectic topology

Let X be a minimal surface of general type with positive geometric genus (b+>1) and let K2 be the square of its canonical class. Building on work of Khodorovskiy and Rana, we prove that if X develops a Wahl singularity of length l in a Q-Gorenstein degeneration, then l≤4K2+7. This improves on the current best-known upper bound due to Lee (l≤400(K2)4). Our bound follows from a stronger theorem constraining symplectic embeddings of certain rational homology balls in surfaces of general type. In particular, we show that if the rational homology ball Bp,1 embeds symplectically in a quintic surface, then p≤12, partially answering the symplectic version of a question of Kronheimer.

Minimal surfaces from infinitesimal deformations of circle packings

We study circle packings with the combinatorics of a triangulated disk in the plane and parametrize deformations of circle packings in terms of vertex rotation and cross ratios. We show that there is a Weierstrass representation formula relating infinitesimal deformations of circle packings to discrete minimal surfaces of Koebe type. Furthermore, every minimal surface of Koebe type can be extended naturally to a discrete minimal surface of general type. In this way, discrete minimal surfaces via Steiner's formula are unified.

Some unlimited families of minimal surfaces of general type with the canonical map of degree 8

In this note, we construct nine families of projective complex minimal surfaces of general type having the canonical map of degree 8 and irregularity 0 or 1. For six of these families the canonical system has a non trivial fixed part.

A new example of an algebraic surface with canonical map of degree 16

In this note, we construct a minimal surface of general type with geometric genus $$ p_g =4 $$ , self-intersection of the canonical divisor $$ K^2 = 32$$ , and irregularity $$ q = 1 $$ such that its canonical map is an Abelian cover of degree 16 of $$\mathbb P^1\times \mathbb P^1$$ .

Examples of Mori dream surfaces of general type with pg = 0

In this paper we study effective, nef and semiample cones of minimal surfaces of general type with pg=0. We provide examples of minimal surfaces of general type with pg=0 and 2≤K2≤9 which are Mori dream spaces. On these examples we also give explicit description of their effective cones with all negative curves. We also present non-minimal surfaces of general type with pg=0 that are not Mori dream surfaces.

Automorphisms of surfaces of general type with q=1 acting trivially in cohomology

Let S be a complex minimal surface of general type with irregularity q(S)=1 and Aut_0(S) the subgroup of automorphisms acting trivially on the cohomology ring with rational coefficients. In this paper we show that |Aut_0(S)|<=4, and if the equality holds then $S$ is a surface isogenous to a product of unmixed type. Moreover, examples of surfaces with |Aut_0(S)|=4 and all possible values of the geometric genus are provided.

On general type surfaces with $$q=1$$ and $$c_2 = 3 p_g$$

Let S be a minimal surface of general type with irregularity $$q(S) = 1$$ . Well-known inequalities between characteristic numbers imply that $$\begin{aligned} 3 p_g(S) \le c_2(S) \le 10 p_g(S), \end{aligned}$$ where $$p_g(S)$$ is the geometric genus and $$c_2(S)$$ the topological Euler characteristic. Surfaces achieving equality for the upper bound are classified, starting with work of Debarre. We study equality in the lower bound, showing that for each $$n \ge 1$$ there exists a surface with $$q = 1$$ , $$p_g = n$$ , and $$c_2 = 3n$$ . The moduli space $$\mathcal {M}_n$$ of such surfaces is a finite set of points, and we prove that $$\#\mathcal {M}_n \rightarrow \infty $$ as $$n \rightarrow \infty $$ . Equivalently, this paper studies the number of closed complex hyperbolic 2-manifolds of first betti number 2 as a function of volume; in particular, such a manifold exists for every possible volume.

On semi‐isogenous mixed surfaces

AbstractLet C be a smooth projective curve and G be a finite subgroup of whose action is mixed, i.e. there are elements in G exchanging the two isotrivial fibrations of . Let be the index two subgroup . If G0 acts freely, then is smooth and we call it semi‐isogenous mixed surface. In this paper we give an algorithm to determine semi‐isogenous mixed surfaces with given geometric genus, irregularity and self‐intersection of the canonical class. As an application we classify irregular semi‐isogenous mixed surfaces with and geometric genus equal to the irregularity; the regular case is subjected to some computational restrictions. In this way we construct new examples of surfaces of general type with . We provide an example of a minimal surface of general type with and .