A Brill–Noether theorem for (toric) surfaces

Brill–Noether theory for cyclic covers

Let [Formula: see text] be the Whitney sum of complex line bundles over a topological space [Formula: see text]. Then, the projectivization [Formula: see text] of [Formula: see text] is called a projective bundle over [Formula: see text]. If [Formula: see text] is a nonsingular complete toric variety, then so is [Formula: see text]. In this paper, we show that the cohomology ring of a nonsingular projective toric variety [Formula: see text] determines whether it admits a projective bundle structure over a nonsingular complete toric surface. In addition, we show that two [Formula: see text]-dimensional projective bundles over [Formula: see text]-dimensional quasitoric manifolds are diffeomorphic if their cohomology rings are isomorphic as graded rings. Furthermore, we study the smooth classification of higher dimensional projective bundles over [Formula: see text]-dimensional quasitoric manifolds.

There is not abstract.

Special divisors on marked chains of cycles

We construct a smooth complex projective rational surface with infinitely many mutually non-isomorphic real forms. This gives the first definite answer to a long-standing open question if a smooth complex projective rational surface has only finitely many non-isomorphic real forms or not.

In this article, we determine the existing condition of cylinders in smooth minimal geometrically rational surfaces over a perfect field. Furthermore, we show that for any birational map between smooth projective surfaces, one contains a cylinder if and only if so does the other.

Let K = k(C) be the function field of an algebraic curve C over an algebraically closed ground field k. Let Γ/K be a smooth projective curve of genus g > 0 with a k-rational point O ∈ Γ(K), and let J/K denote the Jacobian variety of Γ/K. Further let (τ, B) be the K/k:-trace of J (see §2 below and).

We address the following question: Given a polarized toric surface ( S , L ) {(S,{\mathcal{L}})} , and a general integral curve C of geometric genus g in the linear system | L | {|{\mathcal{L}}|} , do there exist degenerations of C in | L | {|{\mathcal{L}}|} to general integral curves of smaller geometric genera? We give an affirmative answer to this question for surfaces associated to h-transverse polygons, provided that the characteristic of the ground field is large enough. We give examples of surfaces in small characteristic, for which the answer to the question is negative. In case the answer is affirmative, we deduce that a general curve C as above is nodal. In characteristic 0, we use the result to show the irreducibility of Severi varieties of a large class of polarized toric surfaces with h-transverse polygon.

We show that a general curve in an explicit class of what we call Du Val pointed curves satisfies the Brill-Noether Theorem for pointed curves. Furthermore, we prove that a generic pencil of Du Val pointed curves is disjoint from all Brill-Noether divisors on the universal curve. This provides explicit examples of smooth pointed curves of arbitrary genus defined over Q which are Brill-Noether general. A similar result is proved for 2-pointed curves as well using explicit curves on elliptic ruled surfaces.

Counterexamples of Kodaira vanishing for smooth surfaces of general type in positive characteristic

Let \(r_d\) be the maximum number of skew lines that a smooth projective surface of degree d (over the complex numbers) can have. It is known that \(r_3=6\), \(r_4=16\) (Schlafli in Q J Math Soc 2:55–65, 110–121, 1858; Nikulin in Math USSR Izv 9:261–275, 1975) and was proven by Miyaoka in 1975 that \(r_d\le 2d(d -2)\) if \(d\ge 4\) (Miyaoka in Math Ann 268:159–172, 1984). Up to now \(r_d\) remains unknown for \(d\ge 5\). However, the lower bound \(d(d-2)+2\) was found by Rams (Proc Am Math Soc 133(1):11–13, 2005) which was improved by Boissiere and Sarti (Ann Scuola Norm Sup Pisa Cl Sci 5:39–52, 2007), who showed that \(d(d-2)+4\le r_d\) for \(d \ge 5\) and odd. In this work, we take the family of degree d smooth surfaces \(\mathcal{R}_d\) in \({\mathbb {P}}^3\) (cf. (1)), considered by Boissiere and Sarti (2007) and study \(r(\mathcal{R}_d)\), the maximum number of skew lines that \(\mathcal{R}_d\) can have. In fact, we prove that \(r(\mathcal{R}_d)\! =\! d(d-2)+4\) if \(d \ge 5\) and odd. Otherwise, we prove that \(r(\mathcal{R}_6)\ge 48\), which implies that Miyaoka’s bound is sharp for \(d=6\), i.e. \(r_6=48\). Still in the even case, we show that \(\mathcal{R}_d\) contains \(d(d-2)+4\) skew lines and we improve the Miyaoka’s bound for the family \(\mathcal{R}_d\) if d is even (Theorem 4.11).

Let P be the projective r-space (r ≥ 3) over an algebraically closed field k. Consider a reduced, irreducible, complete, non-degenerate surface X ⊆ P of degree d, and let \(\bar X\) be its normalization: we shall assume \(\bar X\) to be smooth (which is the case, e. g., when X is the generic projection of a given smooth projective surface \(\bar X\)). Let ∑(n) be the linear system of curves cut out on \(\bar X\) by the hypersurfaces of P of degree n; in other words ∑(n) corresponds to the image of the canonical map $$\rho _n :H^0 (\text{P},O_\text{P} (n)) \to H^0 (\bar X,O_{\bar X} (nD))$$ (where D is the pull-back, via the normalization morphism \(\nu :\bar X \to X\), of the generic hyperplane section X′ of X).

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.

In [19], A. King states the following conjecture: Any smooth complete toric variety has a tilting bundle whose summands are line bundles. The goal of this paper is to prove King’s conjecture for the following types of smooth complete toric varieties: (i) Any d-dimensional smooth complete toric variety with splitting fan. (ii) Any d-dimensional smooth complete toric variety with Picard number ≤2. (iii) The blow up of any smooth complete minimal toric surface at T-invariants points.

Families of Galois closure curves for plane quintic curves