We show that the space $G^r_{\underline d}(X)$ of linear series of certain multi-degree $\underline d=(d_1,d_2)$ (including the balanced ones) and rank r on a general genus-g binary curve X has dimension $\rho _{g,r,d}=g-(r+1)(g-d+r)$ if nonempty, where $d=d_1+d_2$ . This generalizes Caporaso’s result from the case $r\leq 2$ to arbitrary rank, and shows that the space of Osserman-limit linear series on a general binary curve has the expected dimension, which was known for $r\leq 2$ . In addition, we show that the space $G^r_{\underline d}(X)$ is still of expected dimension after imposing certain ramification conditions with respect to a sequence of increasing effective divisors supported on two general points $P_i\in Z_i$ , where $i=1,2$ and $Z_1,Z_2$ are the two components of X. Our result also has potential application to the lifting problem of divisors on graphs to divisors on algebraic curves.

Let $C$ be a Petri general curve of genus $g$ and $E$ a general stable vector bundle of rank $r$ and slope $g-1$ over $C$ with $h^{0}(C,E)=r+1$. For $g\geqslant (2r+2)(2r+1)$, we show how the bundle $E$ can be recovered from the tangent cone to the generalised theta divisor $\unicode[STIX]{x1D6E9}_{E}$ at ${\mathcal{O}}_{C}$. We use this to give a constructive proof and a sharpening of Brivio and Verra’s theorem that the theta map $\mathit{SU}_{C}(r){\dashrightarrow}|r\unicode[STIX]{x1D6E9}|$ is generically injective for large values of $g$.

Let $C$ be a curve over a complete valued field having an infinite residue field and whose skeleton is a chain of loops with generic edge lengths. We prove that any divisor on the chain of loops that is rational over the value group lifts to a divisor of the same rank on $C$, confirming a conjecture of Cools, Draisma, Robeva, and the third author.

Let 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.

Green’s conjecture predicts than one can read off special linear series on an algebraic curve, by looking at the syzygies of its canonical embedding. We extend Voisin’s results on syzygies of K3 sections, to the case of K3 surfaces with arbitrary Picard lattice. This, coupled with results of Voisin and Hirschowitz–Ramanan, provides a complete solution to Green’s conjecture for smooth curves on arbitrary K3 surfaces.