**Edoardo Ballico**

## Curves with Two Pencils and Associated Maps to
Projective Spaces

**Abstract:**

Let $X$ be a smooth and connected projective curve. Assume the existence of
spanned $L\in \mbox{Pic}^a(X)$, $R\in \mbox{Pic}^b(X)$ such that $h^0(X,L) =
h^0(X,R)=2$ and the induced map $\phi _{L,R}:X \to {\bf {P}}^1\times {\bf
{P}}^1$ is birational onto its image. Here we study the following question. What
can be said about the morphisms $\beta :X \to {\bf {P}}^r$ induced by a complete
linear system $\vert L^{\otimes u}\otimes R^{\otimes v}\vert$ for some positive
$u, v$? We study the homogeneous ideal and the minimal free resolution of the
curve $\beta (X)$.

**Keywords:**

Maps to projective spaces, line bundles on curves, curves with two pencils,
homogeneous ideal, minimal free resolution.

**MSC 2000:** 14H51, 14H50