Some reducible and irreducible Brill-Noether loci

1. Problem Statement

The paper investigates the geometry of Brill–Noether loci, denoted $M^r_{g,d}$, which parameterize smooth curves of genus $g$ possessing a linear series of degree $d$ and dimension $r$ (a $g^r_d$).

• Context: For a generic curve, if the Brill–Noether number $\rho(g, r, d) = g - (r+1)(g-d+r)$ is non-negative, the locus is non-empty and irreducible. However, when $\rho < 0$, the generic curve possesses no such series, and the locus forms a proper subvariety of the moduli space $\mathcal{M}_g$.
• Core Question: The authors focus on the reducibility of these loci in the negative $\rho$ range. Specifically, they address:
  1. The existence of curves with specific theta-characteristics (line bundles $L$ such that $L^{\otimes 2} \cong \omega_C$) having a prescribed number of sections ($h^0(L) \ge r+1$).
  2. Whether $M^r_{g,d}$ can have multiple irreducible components, specifically distinguishing between components where the $g^r_d$ is a theta-characteristic and those where it is not.
  3. Establishing optimal bounds for the irreducibility of $M^2_{g,d}$ (plane curves).

2. Methodology

The primary tool employed is the theory of limit linear series on chains of elliptic curves, a technique pioneered by Eisenbud and Harris and refined by the authors and others.

• Limit Linear Series on Chains: The authors construct degenerate curves $X$ consisting of a chain of $g$ elliptic curves $C_1, \dots, C_g$ glued at nodes. A limit linear series on $X$ is defined by a collection of line bundles and sections on each component satisfying vanishing order conditions at the nodes.
• Combinatorial Correspondence: There is a bijection between refined limit linear series and admissible fillings of an $(r+1) \times (g-d+r)$ rectangle with numbers $1, \dots, g$ (Young tableaux).
  • Admissibility Conditions: The fillings must satisfy strict row/column increasing order and specific torsion conditions on the elliptic components (e.g., if an index repeats, the distance between occurrences dictates the torsion order of the nodes).
• Smoothing: The authors utilize criteria (from [31]) to ensure that a limit linear series on a specific chain of elliptic curves (with specific torsion conditions) extends to a linear series on a nearby smooth curve.
• Strategic Construction:
  • To prove the existence of theta-characteristics, they construct symmetric fillings of the rectangle (corresponding to $d=g-1$).
  • To prove reducibility, they construct asymmetric fillings that yield the same dimension but different geometric properties (specifically, the absence of a theta-characteristic).
  • To prove the existence of a third component, they analyze gonality (the existence of a $g^1_k$) and show that certain components contain $k$-gonal curves while others do not.

3. Key Contributions and Results

A. Existence of Theta-Characteristics (Theorem 1)

The authors provide a new, simpler proof of a conjecture by Farkas regarding the existence of theta-characteristics with $r+1$ sections.

• Result: If $g \ge \binom{r+2}{2}$, the locus $T^r_g$ (curves with a theta-characteristic of dimension $r$) is non-empty and has a component of the expected codimension $\binom{r+1}{2}$ in $\mathcal{M}_g$.
• Innovation: Previous proofs used smoothing criteria for curves in projective space. This paper uses limit linear series on chains of elliptic curves. A key technical hurdle was that "naive" symmetric fillings often failed admissibility conditions due to torsion conflicts. The authors resolved this by replacing a pair of elliptic curves in the chain with a single genus 2 curve with specific Weierstrass points, ensuring the limit series smooths correctly.

B. Reducibility of Brill–Noether Loci (Theorem 2)

The paper demonstrates that for $r \ge 3$ and $g = \binom{r+2}{2}$, the locus $M^r_{g, g-1}$ is reducible.

• Two Distinct Components:
  1. $ET^r_g$: A component where the general curve possesses a theta-characteristic (constructed via symmetric fillings).
  2. $N^r_{g, g-1}$: A component where the general curve possesses a $g^r_{g-1}$ that is not a theta-characteristic (constructed via asymmetric fillings).
• Significance: This provides the first examples of reducible Brill–Noether loci for $r \ge 3$ with $d=g-1$. Consequently, this implies the existence of reducible Hilbert schemes for smooth curves of degree $d=g-1$ in $\mathbb{P}^r$.

C. Existence of Three Components (Theorem 3)

For specific values of $r$ ($r=3, 5, 7$ or $r \ge 8$) and $g = \binom{r+2}{2}$, the locus $M^r_{g, g-1}$ has at least three components.

• The Third Component: A component containing the locus of $k$-gonal curves ($M^1_{g, k}$), where $k = \lfloor \frac{r+3}{2} \rfloor$.
• Proof Strategy: The authors show that the components $ET^r_g$ and $N^r_{g, g-1}$ do not contain $k$-gonal curves for small $k$ by analyzing the torsion conditions imposed by their respective fillings (Lemma 4.5). Since the gonality locus has the same or larger dimension, it must constitute a distinct component.

D. Irreducibility for $r=2$ (Proposition 5.2 & Corollary 5.8)

The paper establishes optimal bounds for the irreducibility of $M^2_{g,d}$ (Brill–Noether loci for plane curves).

• Irreducibility Condition: If $\rho(g, 2, d) < 0$ and $d > \frac{g+4}{2}$ (or specific parity cases), $M^2_{g,d}$ is irreducible. The unique component consists of curves with a base-point-free $g^2_d$.
• Reducibility Condition: If $d$ is smaller (specifically $d \le \frac{g+4}{2}$), the locus becomes reducible.
• Mechanism: Reducibility arises when the locus contains a component of expected dimension (from the Severi variety) and a second component of higher dimension coming from curves that factor through a map to a curve of lower genus (specifically $k$-gonal curves where $M^1_{g,k} \subset M^2_{g,d}$).
• Examples: The authors explicitly construct reducible examples for $M^2_{11,7}$, $M^2_{12,8}$, and $M^2_{12,7}$.

4. Significance

• Resolution of Conjectures: The paper confirms Farkas's conjecture on the existence of theta-characteristics with a novel geometric approach.
• Counterexamples to Intuition: It challenges the naive expectation that Brill–Noether loci (and consequently Hilbert schemes) are irreducible when $\rho < 0$. It shows that even in low dimensions ($r=3$) and relatively small genera, these spaces can be highly reducible.
• Hilbert Scheme Implications: The reducibility of $M^r_{g, g-1}$ directly implies that the Hilbert scheme of smooth curves of degree $g-1$ in $\mathbb{P}^r$ is reducible for $r \ge 3$.
• Methodological Advancement: The technique of modifying the chain of elliptic curves (introducing a genus 2 component) to satisfy admissibility conditions for symmetric fillings offers a powerful new tool for constructing limit linear series in cases where standard chains fail.

In summary, Haburcak and Teixidor i Bigas utilize the combinatorial structure of limit linear series on elliptic chains to map out the complex landscape of Brill–Noether loci, proving the existence of multiple components and establishing precise boundaries for irreducibility.