Upper bounds for some Brill–Noether loci over a finite field

We are interested in the quantity ρ ( q , g ) \rho (q,g) defined as the smallest positive integer such that r ≥ ρ ( q , g ) r\geq \rho (q,g) implies that any absolutely irreducible smooth projective algebraic curve defined over F q {\mathbb F}_q of genus g g has a closed point of degree r r . We provide general upper bounds for this number and its exact value for g = 1 , 2 g=1,2 and 3 3 . We also improve the known upper bounds on the number of closed points of degree 2 on a curve.

We construct Abel maps for a stable curve X. Namely, for each oneparameter deformation of X to a smooth curve, having regular total space, and each d ≥ 1, we construct by specialization a map αd X : Ẋd → P d X , where Ẋ ⊆ X is the smooth locus, and P d X is the coarse moduli scheme for equivalence classes of degree-d “semibalanced” line bundles on semistable curves having X as a stable model. For d = 1, we show that α1 X extends to a map α1 X : X → P 1 X , and does not depend on the choice of the deformation. Finally, we give a precise description of when α1 X is injective. The theory of Abel maps for smooth curves goes back to the 19th century. In the modern language, let C be a smooth projective curve, and PicC its degree-d Picard variety parametrizing line bundles of degree d on C. For each d > 0 there exists a remarkable morphism, often called the d-th Abel map: Cd −→ PicC (p1, . . . , pd) 7→ OC( ∑ pi). This map has been extensively studied and used in the literature. For d = 1, after the choice of a “base” point on C, it gives the Abel–Jacobi embedding C ↪→ PicC ∼= PicC (unless C ∼= P1). For an interesting historic survey see [K04] or [K05]. What about Abel maps for singular curves? Abel maps were constructed for all integral curves in [AK], and further studied in [EGK00], [EGK02] and [EK05]. In [AK], it is shown that the first Abel map of an integral singular curve is an embedding into its compactified Picard scheme. However almost nothing is known for reducible curves, not even when they are stable. This lack of knowledge appears all the more regrettable because of the importance of stable curves in moduli theory. In the present paper we construct Abel maps for stable curves. As we see it, Abel maps should satisfy the following natural properties. First, they should have a geometric meaning. More explicitly, recall that for a smooth

Equivariant approach to weighted projective curves

Let M(n, ξ) be the moduli space of stable vector bundles of rank n ≥ 3 and fixed determinant ξ over a complex smooth projective algebraic curve X of genus g ≥ 4. We use the gonality of the curve and r-Hecke morphisms to describe a smooth open set of an irreducible component of the Hilbert scheme of M(n, ξ), and to compute its dimension. We prove similar results for the scheme of morphisms \({M or_P (\mathbb{G}, M(n, \xi))}\) and the moduli space of stable bundles over \({X \times \mathbb{G}}\), where \({\mathbb{G}}\) is the Grassmannian \({\mathbb{G}(n - r, \mathbb{C}^n)}\). Moreover, we give sufficient conditions for \({M or_{2ns}(\mathbb{P}^1, M(n, \xi))}\) to be non-empty, when s ≥ 1.

We show that a principal G-bundle on a smooth projective curve over a finite field is strongly semistable if and only if it is defined by a representation of the fundamental group scheme of the curve into G.

In this paper we construct the Poincare line bundle for the stack of Higgs bundles on smooth projective curves and show that it induces a fully-faithful Fourier-Mukai transform on the category of quasi-coherent sheaves.

According to a theorem of Belyi, a smooth projective algebraic curve is defined over a number field if and only if there exists a non-constant element of its function field ramified only over 0, 1 and ∞. The existence of such a Belyi function is equivalent to that of a representation of the curve as a possibly compactified quotient space of the Poincaré upper half plane by a subgroup of finite index in a Fuchsian triangle group. On the other hand, Fuchsian triangle groups arise in many contexts, such as in the theory of hypergeometric functions and certain triangular billiard problems, which would appear at first sight to have no relation to the Galois problems that motivated the above discovery of Belyi. In this note we review several results related to Belyi's theorem and we develop certain aspects giving examples. For preliminary accounts, see the preprint [Wo1], the conference proceedings article [BauItz] and the “Comptes Rendus” note [CoWo2].

Normal generation of line bundles on multiple coverings

The Brill–Noether theory of curves plays a fundamental role in the theory of curves and their moduli and has been intensively studied since the 19th century. In contrast, Brill–Noether theory for higher dimensional varieties is less understood. It is hard to determine when Brill–Noether loci are nonempty and these loci can be reducible and of larger than the expected dimension. Let $E$ be a semistable sheaf on ${\mathbb{P}}^{2}$. In this paper, we give an upper bound $\beta _{r, \mu }$ for $h^{0}(E)$ in terms of the rank $r$ and the slope $\mu $ of $E$. We show that the bound is achieved precisely when $E$ is a twist of a Steiner bundle. We classify the sheaves $E$ such that $h^{0}(E)$ is within $\lfloor \mu (E) \rfloor + 1$ of $\beta _{r, \mu }$. We determine the nonemptiness, irreducibility and dimension of the Brill–Noether loci in the moduli spaces of sheaves on ${\mathbb{P}}^{2}$ with $h^{0}(E)$ in this range. When they are proper subvarieties, these Brill–Noether loci are irreducible though almost always of larger than the expected dimension.

We produce open subsets of the moduli space of metric graphs without separating edges where the dimensions of Brill–Noether loci are larger than the corresponding Brill–Noether numbers. These graphs also have minimal rank-determining sets that are larger than expected, giving counterexamples to a conjecture of Luo. Furthermore, limits of these graphs have Brill–Noether loci of the expected dimension, so dimensions of Brill–Noether loci of metric graphs do not vary upper semicontinuously in families. Motivated by these examples, we study a notion of rank for the Brill–Noether locus of a metric graph, closely analogous to the Baker–Norine definition of the rank of a divisor. We show that ranks of Brill–Noether loci vary upper semicontinuously in families of metric graphs and are related to dimensions of Brill–Noether loci of algebraic curves by a specialization inequality.

We consider smooth projective curves C/𝔽 over a finite field and their symmetric squares C (2) . For a global function field K/𝔽, we study the K-rational points of C (2) . We describe the adelic points of C (2) surviving Frobenius descent and how the K-rational points fit there. Our methods also lead to an explicit bound on the number of K-rational points of C (2) satisfying an additional condition. Some of our results apply to arbitrary constant subvarieties of abelian varieties, however we produce examples which show that not all of our stronger conclusions extend.

Let [Formula: see text] be a smooth complex projective curve of genus [Formula: see text] and let [Formula: see text] be a line bundle on [Formula: see text] with [Formula: see text]. Let [Formula: see text] be the moduli space of semistable rank 2 [Formula: see text]-twisted Higgs bundles with trivial determinant on [Formula: see text]. Let [Formula: see text] be the moduli space of stable rank 2 [Formula: see text]-twisted Higgs bundles with determinant [Formula: see text] for some [Formula: see text] on [Formula: see text]. We construct a cycle in the product of a stack of rational maps from nonsingular curves to [Formula: see text] and [Formula: see text] by using Hecke modifications of a stable [Formula: see text]-twisted Higgs bundle in [Formula: see text].

A marked Prym curve is a triple $$(C,\alpha ,T_d)$$ ( C , α , T d ) where C is a smooth algebraic curve, $$\alpha $$ α is a $$2-$$ 2 - torsion line bundle on C, and $$T_d$$ T d is a divisor of degree d. We give obstructions—in terms of Gaussian maps—for a marked Prym curve $$(C,\alpha ,T_d)$$ ( C , α , T d ) to admit a singular model lying on an Enriques surface with only one ordinary singular point of multiplicity d, such that $$T_d$$ T d is the pull-back of the singular point by the normalization map. More precisely, let (S, H) be a polarized Enriques surface and let (C, f) be a smooth curve together with a morphism $$f:C \rightarrow S$$ f : C → S birational onto its image and such that $$f(C) \in |H|$$ f ( C ) ∈ | H | , f(C) has exactly one ordinary singular point of multiplicity d. Let $$\alpha =f^*\omega _S$$ α = f ∗ ω S and $$T_d$$ T d be the divisor over the singular point of f(C). We show that if H is sufficiently positive then certain natural Gaussian maps on C, associated with $$\omega _C$$ ω C , $$\alpha $$ α , and $$T_d$$ T d are not surjective. On the contrary, we show that for the general triple in the moduli space of marked Prym curves $$(C,\alpha ,T_d)$$ ( C , α , T d ) , the same Gaussian maps are surjective.

We show that Brill–Noether loci in Hilbert scheme of points on a smooth connected surface $S$ are non-empty whenever their expected dimension is positive and that they are irreducible and have expected dimensions. More precisely, we consider the loci of pairs $(I, s)$, where $I$ is an ideal that locally at the point $s$ of $S$ needs a given number of generators. We give two proofs. The first uses Iarrobino’s description [9] of the Hilbert–Samuel stratification of local punctual Hilbert schemes, and the second is based on induction via birational relationships between different Brill–Noether loci given by nested Hilbert schemes.

Let M be a geometrically irreducible smooth projective variety, defined over a finite field k, such that M admits a k-rational point x0. Let (M,x0/ denote the corresponding fundamental group-scheme introduced by Nori. Let EG be a principal G-bundle over M, where G is a reduced reductive linear algebraic group defined over the field k. Fix a polarization ξ on M. We prove that the following three statements are equivalent:1. The principal G-bundle EG over M is given by a homomorphism (M,x0)→G.2. There are integers b > a ≥ 1, such that the principal G-bundle (FbM)* EG is isomorphic to (FaM) * EG where FM is the absolute Frobenius morphism of M.3. The principal G-bundle EG is strongly semistable, the degree(c2(ad(EG))c1 (ξ)d−2 = 0, where d = dimM, and the degree(c1(EG(χ))c1(ξ)d−1) = 0 for every character χ of G, where EG(χ) is the line bundle over M associated to EG for χ.In [16], the equivalence between the first statement and the third statement was proved under the extra assumption that dimM = 1 and G is semisimple.