I'm looking for some results or references about de Rham cohomology of curves in less-than-optimal cases. The two vanishing results I care about are:

• $H^{k}_{\text{dR}}(X) = 0$ for $k>2$, since $H^{\bullet}_{\text{dR}}$ is a Weil cohomology theory (in characteristic zero), which implies the result if $X$ is a smooth, projective curve over $\text{char}(k)=0$.

• Before taking cohomology, we also have the vanishing $\Omega^i_{X/k}=0$ for $i>1$.

What I would like to know is under what conditions these vanishing statements remain true when allowing one or more of the following hypothesis relaxations:

• Smooth $\to$ singular (or nonsmooth)

• Projective $\to$ quasiprojective

• Characteristic $0$ $\to$ arbitrary characteristic.

Unfortunately I've been unsuccesful in finding proofs or references, or in proving statements myself. So I would appreciate some help.

I'm looking for some results or references about de Rham cohomology of curves in less-than-optimal cases. The two vanishing results I care about are:

• $H^{k}_{\text{dR}}(X) = 0$ for $k>2$, since $H^{\bullet}_{\text{dR}}$ is a Weil cohomology theory (in characteristic zero), which implies the result if $X$ is a smooth, projective curve over $\text{char}(k)=0$.

• Before taking cohomology, we also have the vanishing $\Omega^i_{X/k}=0$ for $i>1$.

What I would like to know is under what conditions these vanishing statements remain true when allowing one or more of the following hypothesis relaxations:

• Smooth $\to$ singular (or nonsmooth) (even reducible, but still connected)

• Projective $\to$ quasiprojective

• Characteristic $0$ $\to$ arbitrary characteristic.

Unfortunately I've been unsuccesful in finding proofs or references, or in proving statements myself. So I would appreciate some help.