Let $\mathbb P^n=\mathbb{CP^n}$ be complex projective space. Let $H^0(\mathcal O_{\mathbb P^n}(d))$ be the group of homogeneous polynomials of degree $d$, and denote its dimension by $N(d)$. Consider the map $$H^0(\mathcal O_{\mathbb P^n}(1))\to Gr(N(d),n)$$ $$L\mapsto H^0(\mathcal O_{\mathbb P^n}(d-1))\cdot L$$ where $Gr$ is the Grassmannian.
My question is
How can I compute the degree of this map?
We can talk about the degree because of Plücker ebeddings.
To give a map to $Gr(V,n)$ one needs to specify a rank $n$ quotient bundle of the trivial bundle $V \otimes O$. In your case $V = H^0(O(d))$, and the quotient bundle is $$ H^0(O(d)) \otimes O \to H^0(O(d-1)) \otimes O(1). $$ Accordingly, if $f : P^n \to Gr$ is the induced map, the pullback of the tautological quotient bundle is $H^0(O(d-1)) \otimes O(1)$, hence the pullback of $O_{Gr}(1)$ is its determinant, which is isomorphic to $O(N(d-1))$. Thus, the degree is $N(d-1)$.