Nonsingular Projective Surface Research Articles

Linearization of transition functions of a semi-positive line bundle along a certain submanifold

Let $X$ be a complex manifold and $L$ be a holomorphic line bundle on $X$. Assume that $L$ is semi-positive, namely $L$ admits a smooth Hermitian metric with semi-positive Chern curvature. Let $Y$ be a compact K\"ahler submanifold of $X$ such that the restriction of $L$ to $Y$ is topologically trivial. We investigate the obstruction for $L$ to be unitary flat on a neighborhood of $Y$ in $X$. As an application, for example, we show the existence of nef, big, and non semi-positive line bundle on a non-singular projective surface.

Higher rank Segre integrals over the Hilbert scheme of points

Let S be a nonsingular projective surface. Each vector bundle V on S of rank s induces a tautological vector bundle over the Hilbert scheme of n points of S . When s=1 , the top Segre classes of the tautological bundles are given by a recently proven formula conjectured in 1999 by M. Lehn. We calculate here the Segre classes of the tautological bundles for all ranks s over all K -trivial surfaces. Furthermore, in rank s=2 , the Segre integrals are determined for all surfaces, thus establishing a full analogue of Lehn's formula. We also give conjectural formulas for certain series of Verlinde Euler characteristics over the Hilbert schemes of points.

Rationality of descendent series for Hilbert and Quot schemes of surfaces

Quot schemes of quotients of a trivial bundle of arbitrary rank on a nonsingular projective surface X carry perfect obstruction theories and virtual fundamental classes whenever the quotient sheaf has at most 1-dimensional support. The associated generating series of virtual Euler characteristics was conjectured to be a rational function in Oprea and Pandharipande (in, Geom Topol. http://arxiv.org/abs/1903.08787 ) when X is simply connected. We conjecture here the rationality of more general descendent series with insertions obtained from the Chern characters of the tautological sheaf. We prove the rationality of descendent series in Hilbert scheme cases for all curve classes and in Quot scheme cases when the curve class is 0.

The virtual [formula omitted]-theory of Quot schemes of surfaces

We study virtual invariants of Quot schemes parametrizing quotients of dimension at most 1 of the trivial sheaf of rank N on nonsingular projective surfaces. We conjecture that the generating series of virtual K-theoretic invariants are given by rational functions. We prove rationality for several geometries including punctual quotients for all surfaces and dimension 1 quotients for surfaces X with pg>0. We also show that the generating series of virtual cobordism classes can be irrational.Given a K-theory class on X of rank r, we associate natural series of virtual Segre and Verlinde numbers. We show that the Segre and Verlinde series match in the following cases: •[(i)] Quot schemes of dimension 0 quotients,•[(ii)] Hilbert schemes of points and curves over surfaces with pg>0,•[(iii)] Quot schemes of minimal elliptic surfaces for quotients supported on fiber classes. Moreover, for punctual quotients of the trivial sheaf of rank N, we prove a new symmetry of the Segre/Verlinde series exchanging r and N. The Segre/Verlinde statements have analogues for punctual Quot schemes over curves.

On the disposition of cubic and pair of conics in a real projective plane

In the first part of the 16th Hilbert problem the question about the topology of nonsingular projective algebraic curves and surfaces was formulated. The problem on topology of algebraic manifolds with singularities belong to this subject too. The particular case of this problem is the study of curves that are decompozable into the product of curves in a general position. This paper deals with the problem of topological classification of mutual positions of a nonsingular curve of degree three and two nonsingular curves of degree two in the real projective plane. Additiolal conditions for this problem include general position of the curves and its maximality; in particular, the number of common points for each pair of curves-factors reaches its maximum. It is proved that the classification contains no more than six specific types of positions of the species under study. Four position types are built, and the question of realizability of the two remaining ones is open.

Nested Hilbert schemes on surfaces: Virtual fundamental class

We construct virtual fundamental classes on nested Hilbert schemes of points and curves in complex nonsingular projective surfaces. These classes recover the virtual classes of Seiberg-Witten theory as well as the (reduced) stable theory, and play a crucial role in the reduced Donaldson-Thomas theory of local-surface-threefolds that we study in [15]. We show that certain integrals against the virtual fundamental classes of punctual nested Hilbert schemes are expressed as integrals over the products of the Hilbert scheme of points. We are able to find explicit formulas for some of these integrals by relating them to Carlsson-Okounkov's vertex operator formulas.

The combinatorics of Lehn's conjecture

Let $S$ be a nonsingular projective surface equipped with a line bundle $H$. Lehn's conjecture is a formula for the top Segre class of the tautological bundle associated to $H$ on the Hilbert scheme of points of $S$. Voisin has recently reduced Lehn's conjecture to the vanishing of certain coefficients of special power series. The first result here is a proof of the vanishings required by Voisin by residue calculations (A. Szenes and M. Vergne have independently found the same proof). Our second result is an elementary solution of the parallel question for the top Segre class on the symmetric power of a nonsingular projective curve $C$ associated to a higher rank vector bundle $V$ on $C$. Finally, we propose a complete conjecture for the top Segre class on the Hilbert scheme of points of $S$ associated to a higher rank vector bundle on $S$ in the $K$-trivial case.

On rational maps from the product of two general curves

This paper treats the dominant rational maps from the product of two very general curves to nonsingular projective surfaces. Combining the result by Bastianelli and Pirola, we prove that the product of two very general curves of genus $g\geq 7$ and $g'\geq 3$ does not admit dominant rational maps of degree $> 1$ if the image surface is non-ruled. We also treat the case of the 2-symmetric product of a curve.

Uniruledness of some moduli spaces of stable pointed curves

We prove uniruledness of some moduli spaces M¯g,n of stable curves of genus g with n marked points using linear systems on nonsingular projective surfaces containing the general curve of genus g. Precisely we show that M¯g,n is uniruled for g=12 and n≤5, g=13 and n≤3, g=15 and n≤2.We then prove that the pointed hyperelliptic locus Hg,n is uniruled for g≥2 and n≤4g+4.In the last part we show that a nonsingular complete intersection surface does not carry a linear system containing the general curve of genus g≥16 and if it carries a linear system containing the general curve of genus 12≤g≤15, then it is canonical.

Arrangements of rational sections over curves and the varieties they define

We introduce arrangements of rational sections over curves. They generalize line arrangements on \mathbb P^2 . Each arrangement of d sections defines a single curve in \mathbb P^{d-2} through the Kapranov's construction of \overline{M}_{0,d+1} . We show a one-to-one correspondence between arrangements of d sections and irreducible curves in M_{0,d+1} , giving also correspondences for two distinguished subclasses: transversal and simple crossing. Then, we associate to each arrangement \mathcal A (and so to each irreducible curve in M_{0,d+1} ) several families of nonsingular projective surfaces X of general type with Chern numbers asymptotically proportional to various log Chern numbers defined by \mathcal A . For example, for the main families and over \mathbb C , any such X is of positive index and \pi_1(X) \simeq \pi_1(\overline{A}) , where \overline{A} is the normalization of A . In this way, any rational curve in M_{0,d+1} produces simply connected surfaces with Chern numbers ratio bigger than 2 . Inequalities like these come from log Chern inequalities, which are in general connected to geometric height inequalities (see Appendix). Along the way, we show examples of étale simply connected surfaces of general type in any characteristic violating any sort of Miyaoka-Yau inequality.

A note on adjoint linear systems

In dimension 2 there is a very useful criterion, due to Miles Reid, giving a necessary and sufficient condition for an adjoint linear system on a nonsingular complex projective surface X to be free at a point \({p\in X}\) , in terms of the coherent cohomology of certain divisors on the blowup of X at p. In this note we extend Miles Reid’s result to higher dimensions.

Minitwistor spaces, Severi varieties, and Einstein–Weyl structure

In this article, we show that the space of nodal rational curves, which is so called a Severi variety (of rational curves), on any non-singular projective surface is always equipped with a natural Einstein–Weyl structure, if the space is 3-dimensional. This is a generalization of the Einstein–Weyl structure on the space of smooth rational curves on a complex surface, given by Hitchin. As geometric objects naturally associated to Einstein–Weyl structure, we investigate null surfaces and geodesics on the Severi varieties. Also, we see that if the projective surface has an appropriate real structure, then the real locus of the Severi variety becomes a positive definite Einstein–Weyl manifold. Moreover, we construct various explicit examples of rational surfaces having 3-dimensional Severi varieties of rational curves.

Canonical fixed parts of fibred algebraic surfaces

It is shown that the fixed part of the canonical linear system of a fibre in a relatively minimal fibred surface supports at most exceptional sets of weakly elliptic singular- ities. Introduction. Let S be a non-singular projective surface and f : S → C a surjective morphism of S onto a non-singular projective curve C with connected fibres. We call f a relatively minimal fibration of genus g if a general fibre is a non-singular projective curve of genus g and there are no (−1)-curves contained in fibres. We assume that g ≥ 2 throughout the paper. Let F be afi bre off . Then the intersection form is negative semi-definite on Supp(F ) by Zariski's lemma. Furthermore, there exist a positive integer m and a 1-connected curve D such that F = mD .W henm is strictly greater than one, F is called a multiple fibre of multiplicity m and OD(D) is a torsion of order m. In (8), we considered the canonical linear system on the minimal resolution of a normal surface singularity and showed that the fixed part supports at most exceptional sets of rational singular points (cf. (1) and (2)). The present article is an extension of it to the semi-global case and we study the fixed part of the canonical linear system |KF | which we call the canonical fixed part in this paper. Recall that the canonical fixed part is closely related to the Horikawa index (see (3, p. 12)), an analytic invariant of a singular fibre germ. In fact, according to (6, Lemma 10 and Theorem 3), if g = 2, the canonical fixed part is a chain of (−2)-curves (of type A) and the Horikawa index is almost equivalent to the number of its irreducible components.

Automorphisms of elliptic surfaces, inducing the identity in cohomology

We give a numerical classification for pairs ( S , G ) consisting of complex nonsingular projective surfaces S of Kodaira dimension one and subgroups G of automorphisms of S inducing trivial actions on H 2 ( S , Q ) .

Arrangements of curves and algebraic surfaces

We prove a strong relation between Chern and log Chern invariants of algebraic surfaces. For a given arrangement of curves, we find nonsingular projective surfaces with Chern ratio arbitrarily close to the log Chern ratio of the log surface defined by the arrangement. Our method is based on sequences of random p p -th root covers, which exploit a certain large scale behavior of Dedekind sums and lengths of continued fractions. We show that randomness is necessary for our asymptotic result, providing another instance of “randomness implies optimal”. As an application over C \mathbb {C} , we construct nonsingular simply connected projective surfaces of general type with large Chern ratio. In particular, we improve the Persson-Peters-Xiao record for Chern ratios of such surfaces.

Flips and variation of moduli scheme of sheaves on a surface

Let $H$ be an ample line bundle on a non-singular projective surface $X$, and $M(H)$ the coarse moduli scheme of rank-two $H$-semistable sheaves with fixed Chern classes on $X$. We show that if $H$ changes and passes through walls to get closer to $K_X$, then $M(H)$ undergoes natural flips with respect to canonical divisors. When $X$ is minimal and $\kappa(X)\geq 1$, this sequence of flips terminates in $M(H_X)$; $H_X$ is an ample line bundle lying so closely to $K_X$ that the canonical divisor of $M(H_X)$ is nef. Remark that so-called Thaddeus-type flips somewhat differ from flips with respect to canonical divisors.

Canonical filtrations and stability of direct images by Frobenius morphisms II

We study the stability of direct images by Frobenius morphisms. We prove that if the cotangent vector bundle of a nonsingular projective surface $X$ is semistable with respect to a numerically positive polarization divisor satisfying certain conditions, then the direct images of the cotangent vector bundle tensored with line bundles on $X$ by Frobenius morphisms are semistable with respect to the polarization. Hence we see that the de Rham complex of $X$ consists of semistable vector bundles if $X$ has the semistable cotangent vector bundle with respect to the polarization with certain mild conditions.

Canonical filtrations and stability of direct images by Frobenius morphisms

We study the stability of direct images by Frobenius morphisms. First, we compute the first Chern classes of direct images of vector bundles by Frobenius morphisms modulo rational equivalence up to torsions. Next, introducing the canonical filtrations, we prove that if $X$ is a nonsingular projective minimal surface of general type with semistable $\Omega_X^1$ with respect to the canonical line bundle $K_X$, then the direct images of line bundles on $X$ by Frobenius morphisms are semistable with respect to $K_X$.

Vanishing of Chern Classes of the De Rham Bundles for Some Families of Moduli Spaces

Given a family of nonsingular complex projective surfaces, there is a corresponding family of Hilbert schemes of zero dimensional subschemes. We prove that the Chern classes, with values in the rational Chow groups, of the de Rham bundles for such a family of Hilbert schemes vanish. A similar result is proved for any relative moduli space of rank one sheaves with trivial integral first Chern class over any family of complex projective surfaces.

AUTOMORPHISMS OF FIBER SURFACES OF GENUS 2, INDUCING THE IDENTITY IN COHOMOLOGY II

We give a complete numerical classification for pairs (S, G) consisting of complex nonsingular projective surfaces S of general type with a fibration of genus 2 and non-trivial subgroups G of automorphisms of S inducing trivial actions on H2(S, ℚ).