Brill Noether Research Articles

Brill–Noether loci of rank 2 vector bundles on a general $\nu $-gonal curve

In this paper we study the Brill Noether locus of rank 2, (semi)stable vector bundles with at least two sections and of suitable degrees on a general ν-gonal curve. We classify its irreducible components having at least expected dimension. We moreover describe the general member F of such components just in terms of extensions of line bundles with suitable minimality properties, providing information on the birational geometry of such components as well as on the very-ampleness of F.

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

Let [Formula: see text] be a smooth projective algebraic curve of genus [Formula: see text], over the finite field [Formula: see text]. A classical result of H. Martens states that the Brill–Noether locus of line bundles [Formula: see text] in [Formula: see text] with [Formula: see text] and [Formula: see text] is of dimension at most [Formula: see text], under conditions that hold when such an [Formula: see text] is both effective and special. We show that the number of such [Formula: see text] that are rational over [Formula: see text] is bounded above by [Formula: see text], with an explicit constant [Formula: see text] that grows exponentially with [Formula: see text]. Our proof uses the Weil estimates for function fields, and is independent of Martens’ theorem. We apply this bound to give a precise lower bound of the form [Formula: see text] for the probability that a line bundle in [Formula: see text] is base point free. This gives an effective version over finite fields of the usual statement that a general line bundle of degree [Formula: see text] is base point free. This is applicable to the author’s work on fast Jacobian group arithmetic for typical divisors on curves.

Higher rank BN-theory for curves of genus 6

Higher rank Brill–Noether theory for genus 6 is especially interesting as, even in the general case, some unexpected phenomena arise which are absent in lower genus. Moreover, it is the first case for which there exist curves of Clifford dimension greater than 1 (smooth plane quintics). In all cases, we obtain new upper bounds for non-emptiness of Brill–Noether loci and construct many examples which approach these upper bounds more closely than those that are well known. Some of our examples of non-empty Brill–Noether loci have negative Brill–Noether numbers.

A Note on Brill–Noether Existence for Graphs of Low Genus

In an influential 2008 paper, Baker proposed a number of conjectures relating the Brill–Noether theory of algebraic curves with a divisor theory on finite graphs. In this note, we examine Baker’s Brill–Noether existence conjecture for special divisors. For g≤5 and ρ(g,r,d) nonnegative, every graph of genus g is shown to admit a divisor of rank r and degree at most d. As further evidence, the conjecture is shown to hold in rank 1 for a number families of highly connected combinatorial types of graphs. In the relevant genera, our arguments give the first combinatorial proof of the Brill–Noether existence theorem for metric graphs, giving a partial answer to a related question of Baker.

Connectedness of Brill–Noether Loci via Degenerations

Abstract We show that limit linear series spaces for chains of curves are reduced. Using recent advances in the foundations of limit linear series, we then use degenerations to study the question of connectedness for spaces of linear series with imposed ramification at up to two points. We find that in general, these spaces may not be connected even when they have positive dimension, but we prove a criterion for connectedness which generalizes the theorem previously proved by Fulton and Lazarsfeld in the case without imposed ramification.

Max Noether's Theorem for integral curves

We study a celebrated result of Max Noether on global sections of the n-dualizing sheaf of a smooth nonhyperelliptic curve in the case where the curve is integral. We reduce the proof of the statement in such a case to a purely numerical condition, which we show that holds if the non-Gorenstein points are bibranch at worst. This is our main result. We also extend the notion of a canonical embedding for integral curves with unibranch non-Gorenstein points at worst, in a way that we can express the dimensions of the components of the ideal in terms of the main invariants of the curve as well. Afterwards we focus on gonality, Clifford index and Koszul cohomology of non-Gorenstein curves by allowing torsion free sheaves of rank 1 in their definitions. We find an upper bound for the gonality, which agrees with Brill–Noether's one for a rational and unibranch curve. We characterize curves of genus 5 with Clifford index 1, and, finally, we study Green's conjecture for a certain class of curves, called nearly Gorenstein.

Some examples of rank-2 Brill\u2013Noether loci

In this paper, we construct some examples of rank-2 Brill–Noether loci with “unexpected” properties on general curves. The key examples are in genus 6, but we also have interesting examples in genus 5 and in higher genus. We relate some of our results to the recent proof of Mercat’s conjecture in rank 2 by Bakker and Farkas.

Interpolation for Brill–Noether space curves

In this note we compute the number of general points through which a general Brill–Noether space curve passes.

Higher Brill–Noether loci and Bogomolov–Gieseker type inequality

We discuss the Brill–Noether loci of the moduli of \(\mu \)-stable sheaves on a smooth projective variety, and obtain some necessary conditions for these loci to be non-empty. As an application of our result, we prove Bogomolov–Gieseker type inequalities concerning the third Chern character of the \(\mu \)-stable sheaves on Calabi–Yau threefolds.

Du Val curves and the pointed Brill–Noether Theorem

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.

Brill–Noether varieties of k-gonal curves

We consider a general curve of fixed gonality k and genus g. We propose an estimate ρ‾g,k(d,r) for the dimension of the variety Wdr(C) of special linear series on C, by solving an analogous problem in tropical geometry. Using work of Coppens and Martens, we prove that this estimate is exactly correct if k≥15g+2, and is an upper bound in all other cases. We also completely characterize the cases in which Wdr(C) has the same dimension as for a general curve of genus g.

Special divisors on marked chains of cycles

We completely describe all Brill–Noether loci on metric graphs consisting of a chain of g cycles with arbitrary edge lengths, generalizing work of Cools, Draisma, Payne, and Robeva. The structure of these loci is determined by displacement tableaux on rectangular partitions, which we define. More generally, we fix a marked point on the rightmost cycle, and completely analyze the loci of divisor classes with specified ramification at the marked point, classifying them using displacement tableaux. Our results give a tropical proof of the generalized Brill–Noether theorem for general marked curves, and serve as a foundation for the analysis of general algebraic curves of fixed gonality.

Brill–Noether theory for cyclic covers

We recall that the Brill–Noether Theorem gives necessary and sufficient conditions for the existence of a gdr. Here we consider a general n-fold, étale, cyclic cover p:C˜→C of a curve C of genus g and investigate for which numbers r,d a gdr exists on C˜. For r=1 this means computing the gonality of C˜. Using degeneration to a special singular example (containing a Castelnuovo canonical curve) and the theory of limit linear series for tree-like curves we show that the Plücker formula yields a necessary condition for the existence of a gdr which is only slightly weaker than the sufficient condition given by the results of Laksov and Kleimann [24], for all n,r,d.

Invariants of the Brill–Noether curve

Abstract For a projective nonsingular curve of genus g, the Brill–Noether locus W d r ( C ) $W^r_d(C)$ parametrizes line bundles of degree d over C with at least r + 1 (linearly independent) sections. When the curve is generic and the Brill–Noether number ρ(g, r, d) equals 1, one can then talk of the Brill–Noether curve. We introduce combinatorial methods that, with help from the theory of limit linear series, allow us to find invariants of Brill–Noether loci. In particular, we explore the first two invariants of the Brill–Noether curve, giving a new way of calculating the genus of this curve and computing its gonality when C has genus 5.

Brill–Noether theory for curves on generic abelian surfaces

We completely describe the Brill-Noether theory for curves in the primitive linear system on generic abelian surfaces, in the following sense: given integers $d$ and $r$, consider the variety $V^r_d(|H|)$ parametrizing curves $C$ in the primitive linear system $|H|$ together with a torsion-free sheaf on $C$ of degree $d$ and $r+1$ global sections. We give a necessary and sufficient condition for this variety to be non-empty, and show that it is either a disjoint union of Grassmannians, or irreducible. Moreover, we show that, when non-empty, it is of expected dimension. This completes prior results by Knutsen, Lelli-Chiesa and Mongardi.

Higher rank BN-theory for curves of genus 4

ABSTRACTHigher rank Brill–Noether theory is completely known for curves of genus ≤ 3. In this paper, we investigate the theory for curves of genus 4. Some of our results apply to curves of arbitrary genus.

Ulrich sheaves and higher-rank Brill–Noether theory

We show that the existence of an Ulrich sheaf on a projective variety X⊆PN is equivalent to the solution of a (possibly) higher-rank Brill–Noether problem for a curve on X that is rarely general in moduli. In addition, we exhibit a large family of curves for which this Brill–Noether problem admits a solution, and we show that existence of an Ulrich sheaf for a finite morphism of smooth projective varieties of any dimension implies sharp numerical constraints involving the degree of the map and the ramification divisor.

The locus of Brill–Noether general graphs is not dense

We provide an example of a trivalent, 3-vertex connected graph G such that, for any choice of metric on G , the resulting metric graph is Brill-Noether special.

Explicit Brill–Noether–Petri general curves

Let p_1,\dots, p_9 be the points in \mathbb A^2(\mathbb Q)\subset \mathbb P^2(\mathbb Q) with coordinates (-2,3),(-1,-4),(2,5),(4,9),(52,375), (5234, 37866),(8, -23), (43, 282), \Bigl(\frac{1}{4}, -\frac{33}{8} \Bigr), respectively. We prove that, for any genus g , a plane curve of degree 3g having a g -tuple point at p_1,\dots, p_8 , and a (g-1) -tuple point at p_9 , and no other singularities, exists and that the general plane curve of that degree and with those singularities is a Brill–Noether–Petri general curve of genus g .

Severi varieties and Brill–Noether theory of curves on abelian surfaces

Abstract Severi varieties and Brill–Noether theory of curves on K3 surfaces are well understood. Yet, quite little is known for curves on abelian surfaces. Given a general abelian surface S with polarization L of type ( 1 , n ) {(1,n)} , we prove nonemptiness and regularity of the Severi variety parametrizing δ-nodal curves in the linear system | L | {|L|} for 0 ≤ δ ≤ n - 1 = p - 2 {0\leq\delta\leq n-1=p-2} (here p is the arithmetic genus of any curve in | L | {|L|} ). We also show that a general genus g curve having as nodal model a hyperplane section of some ( 1 , n ) {(1,n)} -polarized abelian surface admits only finitely many such models up to translation; moreover, any such model lies on finitely many ( 1 , n ) {(1,n)} -polarized abelian surfaces. Under certain assumptions, a conjecture of Dedieu and Sernesi is proved concerning the possibility of deforming a genus g curve in S equigenerically to a nodal curve. The rest of the paper deals with the Brill–Noether theory of curves in | L | {|L|} . It turns out that a general curve in | L | {|L|} is Brill–Noether general. However, as soon as the Brill–Noether number is negative and some other inequalities are satisfied, the locus | L | d r {|L|^{r}_{d}} of smooth curves in | L | {|L|} possessing a g d r {g^{r}_{d}} is nonempty and has a component of the expected dimension. As an application, we obtain the existence of a component of the Brill–Noether locus ℳ p , d r {{\mathcal{M}}^{r}_{p,d}} having the expected codimension in the moduli space of curves ℳ p {{\mathcal{M}}_{p}} . For r = 1 {r=1} , the results are generalized to nodal curves.