Examples of smooth curves of genus $0$ and degree $d>2$.

Question

Can we provide a source of explicit examples ?

The degree assumption $d>2$ means that I would like to see examples which are not conics.

Answer 1

The arithmetic genus of a curve $C\subset \mathbb P^2_k$ of degree $d$ is the number $p_a(C)=dim_k H^1(C, \mathcal O_C)$ and is equal to $\frac {(d-1)(d-2)}{2}$.
Thus that arithmetic genus is $\gt0$ for curves of degree $d\gt2$.
However that curve $C$ has a geometric genus too, equal to the arithmetic genus of the normalization $\tilde C$ : $p_g(C)=p_a(\tilde C)$.
Of course these genera coincide for normal (=nonsingular) curves .

The geometric genus may be zero for curves of degree $d\gt0$ : for example the cusp $C$ given by $y^2z=x^3$ in $\mathbb P^2$ has geometric genus $0$, since its normalization is the projective line $\mathbb P^1$:$$\mathbb P^1=\tilde C\to C:(u:v) \mapsto (x=uv^2:y=v^3: z=u^3) $$

Edit: Addressing the competition
The right honourable user 64494 in his answer below gives the interesting example of the projective plane curve $x^4+x^2yz+yz^3=0$ and reports that his mysterious friend Maple (not a registered user) has computed its geometric genus to be zero.
This is confirmed by our less mysterious friend Julius Plücker, alas deceased (1801-1868), who gave the formula (cf. Seidenberg's book p.129 )$$p_g=\frac {(d-1)(d-2)}{2} -\sum \frac {s_i(s_i-1)}{2} $$ for a plane projective curve having only singularities $P_i$ of multiplicity $s_i$, with distinct tangents .
The curve $x^4+x^2yz+yz^3=0$ has as only singularity $(0:1:0)$, of multiplicity $3$, with three distinct tangents $z=0,x\pm iy=0$. So the displayed formula yields $$p_g=\frac {(4-1)(4-2)}{2} - \frac {3(3-1)}{2} =0 $$ Plücker and Maple embrace each other across the centuries...

Answer 2

A smooth projective plane curve of genus $0$ has to be either a line or a conic.

But the morphism $t \mapsto (t,t^2,t^3,\ldots,t^{d-1})$ (in affine coordinates) embeds $\mathbb P^1$ into $\mathbb P^d$ as a smooth curve of degree $d$. (The image is called a rational normal curve of degree $d$.)

If you choose a generic projection from $\mathbb P^d$ to $\mathbb P^3$, the image of a degree $d$ rational normal curve will be a rational curve of degree $d$ in $\mathbb P^3$.