Let $X$ be a del Pezzo surface (over $\mathbb{C}$), which is obtained by a blow up $\pi: X \rightarrow \mathbb{P}^{2}$ in a collection of points. Let $H$ be the hyperplane class of $\mathbb{P}^{2}$.


**Question**: Consider smooth, irreducible curves $\Sigma$ satisfying the inequality $-K\_{X}\cdot \Sigma > \frac{1}{2}\Sigma \cdot \Sigma + \pi^{\*}H \cdot \Sigma $. 


Is the genus of such curves bounded above?



Curves in del Pezzo surfaces satisfying certain intersection inequality
Here is one proof that the genus is bounded (although presumably a more elegant solution exists).


Write $\Sigma=d\pi^\*H - \sum\_{i=1}^nm\_iE\_i$ in the usual basis for $\operatorname{NS}(X)$ (i.e. $\Sigma$ is the strict transform of a curve of degree $d$ in $\mathbb{P}^2$, with multiplicity $m\_i$ at each of the points which is blown up). 


We can rearrange the inequality into the form $\tfrac12(\Sigma^2 + K\_X\Sigma) < \tfrac12(d-\sum m\_i)$. Then, from the usual genus formula for $\Sigma\subset X$, we have 
\begin{equation\*}
g\_\Sigma = 1 + \tfrac12(\Sigma^2 + K\_X\Sigma) < 1 + \tfrac12\left(d-\sum\_{i=1}^n m\_i\right)
\end{equation\*}
so we may assume that $d > 2 + \sum m\_i$, else $g\_\Sigma \leq 2$.


Now we can prove the result by induction on $n$ (the number of points blown up on $\mathbb{P}^2$). If $n=0$ it is easy to see that the inequality becomes $3d > \tfrac12d^2 + d$, hence $d<4$ and $g\leq1$. 


By the induction hypothesis, if $n>0$ then we may assume each $m\_i\geq 1$ (else blow down the point with $m\_i=0$ and reduce to the $n-1$ case). The inequality is
\begin{equation\*}
3d - \sum\_{i=1}^nm\_i > \frac12 \left(d^2 - \sum\_{i=1}^n m\_i^2 \right) + d
\end{equation\*}
which can be rearranged as
\begin{equation\*}
(d - 2)^2 < \sum\_{i=1}^n (m\_i-1)^2 + 4 - n
\end{equation\*}
But, since $m\_i\geq 1 \: \forall i$, we have $\sum (m\_i-1)^2 \leq \left(\sum(m\_i-1)\right)^2$ and since $\sum m\_i < d - 2$ we have
\begin{equation\*}
(d - 2)^2 < (d-2-n)^2 + 4 - n
\end{equation\*}
which gives the bound $d < \frac{n^2+3n+4}{2n}$, and hence a bound for $g$.


