I want to describe the Gauss map and find its image for the projective variety $V : x^4+y^4+z^4=0$ in $\mathbb{P}^2$.
I know the definition from the book 'An Invitation to Algebraic Geometry' of Springer: $V \rightarrow Gr(d+1,n+1): p \mapsto T_pV$, where $T_pV$ is the projective tangent space to $V$ at the point $p$, and $d = dim(V)$ and $n=2$.
Can someone help me?
This image of the Gauss map of a plane curve is usually referred to as the dual curve.
By Plücker's formula https://en.wikipedia.org/wiki/Pl%C3%BCcker_formula the dual of any smooth quartic curve has degree $12$, $28$ ODP singularities and $24$ cusp singularities. Given these facts, one assumes that the defining equation of the dual curve perhaps isn't anything too familiar.
The process of finding the equations of a dual curve is explained in the Wikipedia article https://en.wikipedia.org/wiki/Dual_curve#Properties_of_dual_curve.
For a point $[p:q:r]$ in the curve, the point $[X:Y:Z]$ in the dual projective space satisfies the four equations:
$$X = \lambda 4 p^3, Y = \lambda 4 q^3 , Z = \lambda 4 r^3, Xp+Yq+Zr = 0.$$
In addition to the original defining equation of the curve. So eliminating the four variables $p,q,r,\lambda$ should allow you to get a defining equation.