Let $\Bbb P^n_q=\Bbb P(q_0,\dotsc, q_n)$ be a weighted projective space and let $f_1,\dotsc, f_k$ be $q$-homogeneous polynomials of degrees $d_1,\dotsc,d_k$ respectively. Let $X$ be defined as the intersection of the hypersurfaces in $\Bbb P^n_q$ defined by $f_1,\dotsc,f_k$.

Is there a criterion to test whether or not $X$ is smooth?

If $q=(1,\dotsc,1)$, then $\Bbb P^n_q$ is ordinary projective space and we can cover $\Bbb P^n$ with affine spaces and use the Jacobian criterion. This process fails, however, in general since $\Bbb P^n_q$ is singular in general.


There is the concept of quasi-smooth subvarieties that have many properties as smooth subvarieties of smooth projective spaces.

Take a look at Dolgachev's Weighted Projective Varieties.

You can also analyse the situation on a orbifold chart around a point.


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