Elliptic curve(i.e. Compact Riemann surfaces with genus one) $\Sigma$ can be described by an equation $y^2-x(x-1)(x-\lambda)=0, \lambda\neq 0,1$. In homogeneous coordinates $(x,y,z)$, the equation becomes $$y^2z-x^3+(1+\lambda)x^2z-\lambda xz^2=0.$$ We should check the curve in $\mathbb{P}^2$ is everywhere non-sigular, so we check the partial derivatives of $P(x,y,z)=y^2z-x^3+(1+\lambda)x^2z-\lambda xz^2$ do not all vanish at any point of $\Sigma$.
My question is that could I conclude that $\Sigma$ is compact if the partial derivatives of $P(x,y,z)$ do not all vanish at any point of $\Sigma$?
Your two conditions have no relations to each other. $\Sigma(\Bbb C)$ is compact because it is a closed subset of a compact space. Checking that the partial derivatives do not all vanish at some point in $\Sigma$ gives that $\Sigma$ is smooth, or that $\Sigma(\Bbb C)$ is a complex manifold.