Brill Noether Loci Research Articles

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.

Higher rank BN-theory for curves of genus 5

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.

Deformations of minimal cohomology classes on abelian varieties

We show that the infinitesimal deformations of Brill–Noether loci [Formula: see text] attached to a smooth non-hyperelliptic curve [Formula: see text] are in one-to-ne correspondence with the deformations of [Formula: see text]. As an application, we prove that if a Jacobian [Formula: see text] deforms together with a minimal cohomology class out the Jacobian locus, then [Formula: see text] is hyperelliptic. In particular, this provides an evidence toward a conjecture of Debarre on the classification of ppavs carrying a minimal cohomology class.

Theta-regularity of curves and Brill–Noether loci

We provide a bound on the $\Theta$-regularity of an arbitrary reduced and irreducible curve embedded in a polarized abelian variety in terms of its degree and codimension. This is an "abelian" version of Gruson-Lazarsfeld-Peskine's bound on the Castelnuovo--Mumford regularity of a non-degenerate curve embedded in a projective space. As an application, we provide a Castelnuovo type bound for the genus of a curve in a (non necessarily principally) polarized abelian variety. Finally, we bound the $\Theta$-regularity of a class of higher dimensional subvarieties in Jacobian varieties, i.e. the Brill-Noether loci associated to a Petri general curve, extending earlier work of Pareschi-Popa.

Nonemptiness of Brill–Noether Loci in M(2, K)

Let C be a smooth projective complex curve of genus g ≥ 2. We investigate the Brill–Noether locus consisting of stable bundles of rank 2 and canonical determinant having at least k independent sections. Using the Hecke correspondence, we construct a fundamental class, which determines the nonemptiness of this locus at least when C is a Petri curve. We prove that in many expected cases the Brill–Noether locus is nonempty. For some values of k, the result is best possible.

Nonemptiness of Brill–Noether loci in M(2,L)

Let [Formula: see text] be a smooth projective complex curve of genus [Formula: see text]. We investigate the Brill–Noether locus consisting of stable bundles of rank 2 and determinant [Formula: see text] of odd degree [Formula: see text] having at least [Formula: see text] independent sections. This locus possesses a virtual fundamental class. We show that in many cases this class is nonzero, which implies that the Brill–Noether locus is nonempty. For many values of [Formula: see text] and [Formula: see text] the result is best possible. We obtain more precise results for [Formula: see text]. Appendix A contains the proof of a combinatorial lemma which we need.

Linked determinantal loci and limit linear series

We study (a generalization of) the notion of linked determinantal loci recently introduced by the second author, showing that as with classical determinantal loci, they are Cohen-Macaulay whenever they have the expected codimension. We apply this to prove Cohen-Macaulayness and flatness for moduli spaces of limit linear series, and to prove a comparison result between the scheme structures of Eisenbud-Harris limit linear series and the spaces of limit linear series recently constructed by the second author. This comparison result is crucial in order to study the geometry of Brill-Noether loci via degenerations.

Linked symplectic forms and limit linear series in rank 2 with special determinant

We generalize the prior linked symplectic Grassmannian construction, applying it to prove smoothing results for rank-2 limit linear series with fixed special determinant on chains of curves. We apply this general machinery to prove new results on nonemptiness and dimension of rank-2 Brill–Noether loci in a range of degrees.

The osculating cone to special Brill–Noether loci

We describe the osculating cone to Brill–Noether loci $$W^0_d(C)$$ at smooth isolated points of $$W^1_d(C)$$ for a general canonically embedded curve C of even genus $$g=2(d-1)$$ . In particular, we show that the canonical curve C is a component of the osculating cone. The proof is based on techniques introduced by George Kempf.

Brill–Noether Locus of Rank 1 and Degree g− 1 on a Nodal Curve

In this paper we consider the Brill–Noether locus of line bundles of multidegree of total degree g − 1 having a nonzero section over a connected nodal curve C of genus g. We give an explicit description of the irreducible components of for a semistable multidegree . As a consequence we show that, if two semistable multidegrees of total degree g − 1 over a curve with no rational components differ by a twister, then the corresponding Brill–Noether loci are isomorphic.

Rank two bundles with canonical determinant and four sections

Let \(C\) be a smooth irreducible complex projective curve of genus \(g\) and let \(B^k(2,K_C)\) be the Brill–Noether locus parametrizing classes of (semi)-stable vector bundles \(E\) of rank two with canonical determinant over \(C\) with \(h^0(C,E)\ge k\). We show that \(B^4(2,K_C)\) has an irreducible component \(\mathcal B\) of dimension \(3g-13\) on a general curve \(C\) of genus \(g\ge 8\). Moreover, we show that for the general element \([E]\) of \(\mathcal B\), \(E\) fits into an exact sequence \(0\rightarrow {\mathcal {O}}_C(D)\rightarrow E\rightarrow K_C(-D)\rightarrow 0\) with \(D\) a general effective divisor of degree three, and the corresponding coboundary map \(\partial : H^0(C,K_C(-D))\rightarrow H^1(C,{\mathcal {O}}_C(D))\) has cokernel of dimension three.

Brill–Noether loci for divisors on irregular varieties

We take up the study of the Brill-Noether loci W^r(L,X):=\{\eta\in \mathrm {Pic}^0(X)\ |\ h^0(L\otimes\eta)\ge r+1\} , where X is a smooth projective variety of dimension >1 , L\in \mathrm {Pic}(X) , and r\ge 0 is an integer. By studying the infinitesimal structure of these loci and the Petri map (defined in analogy with the case of curves), we obtain lower bounds for h^0(K_D) , where D is a divisor that moves linearly on a smooth projective variety X of maximal Albanese dimension. In this way we sharpen the results of [Xi] and we generalize them to dimension >2 . In the 2 -dimensional case we prove an existence theorem: we define a Brill-Noether number \rho(C, r) for a curve C on a smooth surface X of maximal Albanese dimension and we prove, under some mild additional assumptions, that if \rho(C, r)\ge 0 then W^r(C,X) is nonempty of dimension \ge \rho(C,r) . Inequalities for the numerical invariants of curves that do not move linearly on a surface of maximal Albanese dimension are obtained as an application of the previous results.

Injectivity of the Petri map for twisted Brill–Noether loci

Let C be a generic curve, E a generic vector bundle on C. Then, for every line bundle on C the twisted Petri map $$P_{E}:H^0(C,L\otimes E)\otimes H^0(C, K\otimes L^*\otimes E^{*})\rightarrow H^0(C, K)$$ is injective.

The Brill–Noether rank of a tropical curve

We construct a space classifying divisor classes of a fixed degree on all tropical curves of a fixed combinatorial type and show that the function taking a divisor class to its rank is upper semicontinuous. We extend the definition of the Brill-Noether rank of a metric graph to tropical curves and use the upper semicontinuity of the rank function on divisors to show that the Brill-Noether rank varies upper semicontinuously in families of tropical curves. Furthermore, we present a specialization lemma relating the Brill-Noether rank of a tropical curve with the dimension of the Brill-Noether locus of an algerbaic curve.

Abelian varieties in Brill–Noether loci

In this paper, improving on results of Abramovich, Harris, Debarre and Fahlaoui [1,8], we give the full classification of curves C of genus g such that a Brill–Noether locus Wds(C), strictly contained in the jacobian J(C) of C, contains a variety Z stable under translations by the elements of a positive dimensional abelian subvariety A⊊J(C) and such that dim(Z)=d−dim(A)−2s, i.e., the maximum possible for such a Z.

Vector bundles on Fano threefolds of genus 7 and Brill–Noether loci

Let X be a smooth prime Fano threefold of genus 7 and let Γ be its homologically projectively dual curve. We prove that, for d ≥ 6, an irreducible component of the moduli scheme MX(2, 1, d) of rank-2 stable sheaves on X with c1= 1, c2= d is birational to a generically smooth (2d - 9)-dimensional component of the Brill–Noether variety [Formula: see text] of stable vector bundles on Γ of rank d - 5 and degree 5d-24 with at least 2d - 10 independent global sections. The space MX(2, 1, 6) is proved to be isomorphic to [Formula: see text], and to be a smooth irreducible threefold if X is general enough.

BRILL–NOETHER LOCI WITH FIXED DETERMINANT IN RANK 2

In the 1990s, Bertram, Feinberg and Mukai examined Brill–Noether loci for vector bundles of rank 2 with fixed canonical determinant, noting that the dimension was always bigger in this case than the naive expectation. We generalize their results to treat a much broader range of fixed-determinant Brill–Noether loci. The main technique is a careful study of symplectic Grassmannians and related concepts.

SPECIAL DETERMINANTS IN HIGHER-RANK BRILL–NOETHER THEORY

Extending our previous study of modified expected dimensions for rank-2 Brill–Noether loci with prescribed special determinants, we introduce a general framework which applies a priori for arbitrary rank, and use it to prove modified expected dimension bounds in several new cases, applying both to rank 2 and to higher rank. The main tool is the introduction of generalized alternating Grassmannians, which are the loci inside Grassmannians corresponding to subspaces which are simultaneously isotropic for a family of multilinear alternating forms on the ambient vector space. In the case of rank 2 with two-dimensional spaces of sections, we adapt arguments due to Teixidor i Bigas to show that our new modified expected dimensions are in fact sharp.

Brill–Noether loci in codimension two

AbstractLet us consider the locus in the moduli space of curves of genus$2k$defined by curves with a pencil of degree$k$. Since the Brill–Noether number is equal to$- 2$, such a locus has codimension two. Using the method of test surfaces, we compute the class of its closure in the moduli space of stable curves.

Maps into projective spaces

We compute the cohomology of the Picard bundle on the desingularization \(\tilde{J}^d(Y)\) of the compactified Jacobian of an irreducible nodal curve Y. We use it to compute the cohomology classes of the Brill–Noether loci in \(\tilde{J}^d(Y)\).