On the Moduli Space of Coherent Systems of Type $(2, c_1, c_2, 2)$ on Projective Plane

We study the moduli space of coherent systems in $P^2$ using the Segre invariant. We obtain necessary conditions for the existence of $\alpha$-semistable coherent systems $(E,V)$ of type $(2, c_1, c_2, k)$, with $k \geq 2$. Afterwards, we give numerical conditions to the nonemptiness of the moduli space and compute the critical values depending of the Chern classes. Finally, we give some topological properties of the flips.