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I'm not sure how one deals with the general case, but your example is the cone in $\mathbb P^{N+1}$ over the $k$-th Veronese embedding of $\mathbb P^{n-1}(\mathbb K)$ in $\mathbb P^N$ ($n=$ number of 1's, $N=$ the appropriate dimension for the Veronese embedding).

So for $k>1$ the automorphism group $G$ is an extension $$1\to \mathbb K^*\times {\mathbb K}^{N+1}\to G\to \mathbb PGL(n)\to 1.$$ This is easily seen by taking homogeneous coordinates in $\mathbb P^{N+1}$ such that the vertex of the cone is, say, $[1,0\dots 0]$ and representing an automorphism by a matrix such that the first column is $^t(1,0\dots 0)$.

I'm not sure how one deals with the general case, but your example is the cone in $\mathbb P^{N+1}$ over the $k$-th Veronese embedding of $\mathbb P^{n-1}(\mathbb K)$ in $\mathbb P^N$ ($n=$ number of 1's, $N=$ the appropriate dimension for the Veronese embedding).

So for $k>1$ the automorphism group $G$ is an extension $$1\to \mathbb K^*\times {\mathbb K}^{N+1}\to G\to \mathbb PGL(n)\to 1.$$ EDIT: as J'er'emy points out below and in his answer, the kernel in the sequence above is not a direct product, but a more complicated group.

This is easily seen by taking homogeneous coordinates in $\mathbb P^{N+1}$ such that the vertex of the cone is, say, $[1,0\dots 0]$ and representing an automorphism by a matrix such that the first column is $^t(1,0\dots 0)$.

I'm not sure how one deals with the general case, but your example is the cone in $\mathbb P^{N+1}$ over the $k$-th Veronese embedding of $\mathbb P^{n-1}(\mathbb K)$ in $\mathbb P^N$ ($n=$ number of 1's, $N=$ the appropriate dimension for the Veronese embedding).

So the automorphism group $G$ is an extension $$1\to \mathbb K^*\times {\mathbb K}^{N+1}\to G\to \mathbb PGL(n)\to 1.$$ This is easily seen by taking homogeneous coordinates in $\mathbb P^{N+1}$ such that the vertex of the cone is, say, $[1,0\dots 0]$ and representing an automorphism by a matrix such that the first column is $^t(1,0\dots 0)$.

I'm not sure how one deals with the general case, but your example is the cone in $\mathbb P^{N+1}$ over the $k$-th Veronese embedding of $\mathbb P^{n-1}(\mathbb K)$ in $\mathbb P^N$ ($n=$ number of 1's, $N=$ the appropriate dimension for the Veronese embedding).

So for $k>1$ the automorphism group $G$ is an extension $$1\to \mathbb K^*\times {\mathbb K}^{N+1}\to G\to \mathbb PGL(n)\to 1.$$ This is easily seen by taking homogeneous coordinates in $\mathbb P^{N+1}$ such that the vertex of the cone is, say, $[1,0\dots 0]$ and representing an automorphism by a matrix such that the first column is $^t(1,0\dots 0)$.

I'm not sure how one deals with the general case, but your example is the cone in $\mathbb P^{n+1}$ over the $k$-th Veronese embedding of $\mathbb P^{n-1}(\mathbb K)$ in $\mathbb P^n$ ($n=$ number of 1's, $N=$ the appropriate dimension for the Veronese embedding).

So the automorphism group $G$ is an extension $$1\to \mathbb K^*\times {\mathbb K}^{N+1}\to G\to \mathbb PGL(n)\to 1.$$ This is easily seen by taking homogeneous coordinates in $\mathbb P^{N+1}$ such that the vertex of the cone is, say, $[1,0\dots 0]$.

I'm not sure how one deals with the general case, but your example is the cone in $\mathbb P^{N+1}$ over the $k$-th Veronese embedding of $\mathbb P^{n-1}(\mathbb K)$ in $\mathbb P^N$ ($n=$ number of 1's, $N=$ the appropriate dimension for the Veronese embedding).

So the automorphism group $G$ is an extension $$1\to \mathbb K^*\times {\mathbb K}^{N+1}\to G\to \mathbb PGL(n)\to 1.$$ This is easily seen by taking homogeneous coordinates in $\mathbb P^{N+1}$ such that the vertex of the cone is, say, $[1,0\dots 0]$ and representing an automorphism by a matrix such that the first column is $^t(1,0\dots 0)$.