Description
I will go into some detail into the proof of the main ingredients of a recent important result on planar arcs by Ball and Lavrauw:
Assume $A$ is an arc in $PG(2,q)$ that is not contained in a conic.
If $A$ is not contained in a curve of deg $t$, then it is contained in the intersection of two curves of degree at most $t+p^{\lfloor \log_p t\rfloor}$ not sharing a common component.
If $A$ is contained in a curve of deg $t$ and $p^{\lfloor \log_p t\rfloor)(t+\frac12p^{\lfloor \log_p t\rfloor}+\frac32)\le \frac12(t+2)(t+1)$ then there is another curve of deg at most $t+p^{\lfloor \log_p t\rfloor}$ containing $A$ and sharing no common component with $\phi$.
Important consequences of this are: $A$ is contained in a conic if
(i) $q$ is an odd square, and $|A|\ge q-\sqrt q+3+\sqrt{q}/p$, or
(ii) $q$ prime, $|A|\ge q-\sqrt q+\frac 72$.