= Standard monomials of the twisted cubic
{title2=$1,\ x_1,\ x_2\text{ over }k[x_0,x_3]$}
Modulo the three quadric relations of the <twisted cubic>, replace $x_1^2$ by $x_0x_2$, $x_2^2$ by $x_1x_3$, and $x_1x_2$ by $x_0x_3$. Each replacement reduces the total exponent of $x_1,x_2$. The resulting monomials are $x_0^ax_3^b$, $x_0^ax_1x_3^b$, and $x_0^ax_2x_3^b$. In degree $m$ their images under $x_i\mapsto s^{3-i}t^i$ are all $3m+1$ distinct monomials of degree $3m$. They are linearly independent, so the three quadrics generate the full <homogeneous ideal>, not merely its radical. The quotient is the third <Veronese subring> of $k[s,t]$.
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