Existence of pencils on some special curves of genus 10 Everything over $\Bbb{C}$. Say we have a smooth curve $C$ of genus $10$ which is a double cover of a smooth plane cubic curve. Therefore $C$ admits a 1-dimensional family of pencils of degree 4 (arising from the involutions on the cubic).
Can we deduce from this that $C$ does not admit any pencil of degree $3$ ? (in other words that $W^1_3( C )$ is empty)
 A: For every integer $p_a>4$, there does not exist a smooth, projective, geometrically connected curve of genus $p_a$ that admits both a degree $2$, finite, flat morphism, $f:C\to E$, to a smooth plane cubic $E$ and a degree $3$, finite, flat morphism, $g:C\to \mathbb{P}^1$, to the projective line.  Probably this can be proved directly from the geometric Riemann-Roch theorem, but the argument below just uses the adjunction formula.  Note also, a complete intersection in $\mathbb{P}^2\times \mathbb{P}^1$ of hypersurfaces of bidegrees $(3,0)$ and $(1,2)$ is a curve of arithmetic genus $p_a=4$ that does admit a degree $2$ morphism to $E$ and a degree $3$ morphism to $\mathbb{P}^1$.  So the inequality $p_a>4$ is necessary.
By way of contradiction, assume that there is such a curve.  Consider the product morphism, $$(f,g):C\to E\times\mathbb{P}^1.$$  Denote the image Cartier divisor by $$i:B\to E\times \mathbb{P}^1.$$  This is a closed, integral (possibly singular) curve in $E\times \mathbb{P}^1$.  Denote by $$h:C\to B,$$ the unique finite, surjective morphism such that $i\circ h$ equals $(f,g)$.  
What is the degree of $h$?  For every invertible $\mathcal{O}_E$-module $\mathcal{A}$ of degree $1$, the invertible $\mathcal{O}_C$-module $h^*(i^*\text{pr}_E^*\mathcal{A})$ has degree $2$.  Thus, the degree of $h$ divides $2$.  Similarly, the degree of $h^*(i^*\text{pr}_{\mathbb{P}^1}^*\mathcal{O}(1))$ has degree $3$.  Thus, the degree of $h$ also divides $3$.  Therefore the degree of $h$ equals $1$, i.e., $h$ is a birational equivalence.  In particular, the arithmetic genus of $B$ is at least as large as the geometric genus $2p_a-2 > 6$ of $C$.
Because $\text{Pic}(E\times \mathbb{P}^1)$ equals $\text{Pic}(E)\times \mathbb{Z}$,  there is an isomorphism of invertible sheaves $$\mathcal{O}_{E\times \mathbb{P}^1}(\underline{B}) \cong \text{pr}_E^*\mathcal{L}\otimes \text{pr}_{\mathbb{P}^1}^*\mathcal{O}(d),$$  for an invertible sheaf $\mathcal{L}$ on $E$ and for an integer $d$.  By the computations above, $d$ must equal $2$ and $\mathcal{L}$ must have degree $3$.  By the adjunction formula, there is an isomorphism of invertible sheaves, $$\omega_B \cong \left( \text{pr}_E^*\mathcal{L} \right)|_B.$$  The degree of this invertible sheaf on $B$ equals $6$, which is strictly less than $2p_a-2$.
Edit. Let $E$ and $F$ be smooth projective curves such that the natural map $\text{Pic}(E)\times \text{Pic}(F) \to \text{Pic}(E\times F)$ is an isomorphism.  Let $C$ be a smooth, projective curve.  Let $\epsilon:C\to E$ and $\phi:C\to F$ be finite, flat morphisms such that the product morphism $(\epsilon,\phi):C\to E\times F$ is birational to its image, e.g., this holds if $\text{deg}(\epsilon)$ and $\text{deg}(\phi)$ are relatively prime.  Then the same argument as above implies the following inequality, $$p_a(C) \leq (\text{deg}(\epsilon)-1)(\text{deg}(\phi)-1) + p_a(E)\text{deg}(\epsilon) + p_a(F)\text{deg}(\phi). $$
In particular, when $F$ is $\mathbb{P}^1$ and $d$ denotes $\text{deg}(\phi)$, so that we are considering $g^1_d$s on $C$, this reduces to the following inequality, $$d \geq \frac{(2p_a(C)-2) - \text{deg}(\epsilon)(2p_a(E)-2)}{2(\text{deg}(\epsilon)-1)} = \frac{\text{deg}(\text{Branch}(\epsilon))}{2(\text{deg}(\epsilon)-1)}.$$  Finally, if also $E$ has genus $1$, this becomes the following, $$d \geq \frac{p_a(C)-1}{\text{deg}(\epsilon)-1}.$$
Second Edit. Thanks to Felipe Voloch who recognized this inequality.  It is the Castelnuovo-Severi inequality.
