Pencils and nets on curves arising from rank 1 sheaves on K3 surfaces

Let $S$ be a K3 surface, $C$ a smooth curve on $S$ with $\mathcal{O}_S(C)$ ample, and $A$ a base-point free $g^2_d$ on $C$ of small degree. We use Lazarsfeld--Mukai bundles to prove that $A$ is cut out by the global sections of a rank 1 torsion-free sheaf $\mathcal{G}$ on $S$. Furthermore, we show that $c_1(\mathcal{G})$ with one exception is adapted to $\mathcal{O}_S(C)$ and satisfies $\mathrm{Cliff}(c_1(\mathcal{G})_{|C})\leq\mathrm{Cliff}(A)$, thereby confirming a conjecture posed by Donagi and Morrison. We also show that the same methods can be used to give a simple proof of the conjecture in the $g^1_d$ case. In the final section, we give an example of the mentioned exception where $h^0(C,c_1(\mathcal{G})_{|C})$ is dependent on the curve $C$ in its linear system, thereby failing to be adapted to $\mathcal{O}_S(C)$.Comment: 11 pages. Published versio

Rank-width of Random Graphs

Rank-width of a graph G, denoted by rw(G), is a width parameter of graphs introduced by Oum and Seymour (2006). We investigate the asymptotic behavior of rank-width of a random graph G(n,p). We show that, asymptotically almost surely, (i) if 0<p<1 is a constant, then rw(G(n,p)) = \lceil n/3 \rceil-O(1), (ii) if 1/n<< p <1/2, then rw(G(n,p))= \lceil n/3\rceil-o(n), (iii) if p = c/n and c > 1, then rw(G(n,p)) > r n for some r = r(c), and (iv) if p <= c/n and c<1, then rw(G(n,p)) <=2. As a corollary, we deduce that G(n,p) has linear tree-width whenever p=c/n for each c>1, answering a question of Gao (2006).Comment: 10 page

Chowla's cosine problem

Suppose that G is a discrete abelian group and A is a finite symmetric subset of G. We show two main results. i) Either there is a set H of O(log^c|A|) subgroups of G with |A \triangle \bigcup H| = o(|A|), or there is a character X on G such that -wh{1_A}(X) >> log^c|A|. ii) If G is finite and |A|>> |G| then either there is a subgroup H of G such that |A \triangle H| = o(|A|), or there is a character X on G such that -wh{1_A}(X)>> |A|^c.Comment: 21 pp. Corrected typos. Minor revision