Which weighted projective spaces (and their finite quotients) are local complete intersections?

Question

Let $G$ be a finite subgroup of $\textrm{Gl}_{n+1}(k)$ (where $k$ is an algebraically closed field). My question is: do there exist examples of $G$ such that the corresponding quotient $P$ of $\mathbb{P}^n$ by $G$ is not isomorphic to $\mathbb{P}^n$ but yet is locally a set-theoretic complete intersection (if we embed it into $\mathbb{P}^m$ for some $m>n$)? It seems that the most "popular" quotients of this type are the so-called weighted projective spaces; so what is known for them? Can $P$ be smooth, and still not isomorphic to $\mathbb{P}^n$?

I am also interested in cases when $P$ can locally be "described by not so many extra equations" (say, by $m-3/4 n$ for $P\subset \mathbb{P}^m$) and when the "non-LSTCI locus" (i.e., the set of points where $m-n$ equations are not sufficient) is of dimension less than $n/4$.

Upd. I am deeply grateful to Francesco Polizzi for his answer (including examples). So I would like to ask the following: can one describe weighted projective spaces (or more general $P$ as above) that are locally (set-theoretic) complete interestions? Is it possible to calculate the number of "extra local equations" for weighted projective spaces? Any references (and electronic versions of these text) would also be very welcome!

Answer 1

Regarding your question about weighted projective spaces, a lot is known about them, see for instance [1] and [2].

In particular, any weighted projective space $\mathbb{P}(\mathcal Q)$ is irreducible, normal, Cohen-Macaulay and has at most cyclic quotient singularities (hence rational singularities), see [2], p. 122.

However, weighted projective spaces are not locally complete intersections (l.c.i.) in general. For instance, the weighted projective plane $\mathbb{P}(1, \, 1, \, 2)$ is isomorphic to a quadric cone, so it is a l.c.i. projective variety, whereas $\mathbb{P}(1, \, 1, \,3)$ is isomorphic to a cone over a twisted cubic, hence it has a $\frac{1}{3}(1, \, 1)$-singularity at its vertex, which is not a complete intersection singularity. More generally, $\mathbb{P}(1, \, 1, \,n)$ is a cone over a rational normal curve of degree $n$, so it is never a l.c.i. variety for $n \geq 3$.

Furthermore, any smooth $\mathbb{P}(\mathcal Q)$ is isomorphic to $\mathbb{P}^r$, see [1], p. 39

References.

[1] I. Dolgachev: Weighted projective varieties, Group actions and vector fields, Lecture Notes in Math. 956 (1982), 34-71.

[2] M. Beltrametti, L. Robbiano: Introduction to the theory of weighted projective spaces, Expo. Math. 4 (1986), 111-162.

Answer 2

Weighted projective spaces (WPS) are toric varieties of Picard number one. In general, any complete $\mathbb{Q}$-factorial toric variety of Picard number one is a "Fake weighted projective space" and it is obtained as a finite abelian quotient of a WPS as follows :

Let $X$ be a $\mathbb{Q}$-factorial complete toric variety of Picard number one and dimension $n$. Then, its fan is composed by exactly $n+1$ simplicial cones of maximal dimension. These cones are generated by $n+1$ primitive lattice vectors $v_0,\ldots,v_n\in N \subseteq N_\mathbb{R}\cong \mathbb{R}^n$. As they are linearly dependent, they satisfy the relation

$$\sum_{i=0}^n a_i v_i = 0. $$

Let $N'\subseteq N$ be the sublattice generated by the vectors $v_0,\ldots, v_n$. Then, $N/N'$ is a finite abelian group (isomorphic to the torsion group of $\operatorname{Cl}(X)$, which is also isomorphic to the fondamental group in codimension 1 of $X$) inducing a quotient map

$$ \mathbb{P}(a_0,\ldots,a_n) \to X $$

Now, if you want to know which of them are local complete intersections you might refer to Nakajima's classification.

I've never seen the computation Nakajima's conditions in the context of WPS (as far as I know), but it should be doable (?).

Some references:

1. W. Buczynska, "Fake weighted projective spaces". http://arxiv.org/abs/0805.1211

2. M. Rossi, L. Terracini, "A Q-factorial complete toric variety is a quotient of a poly weighted space". http://arxiv.org/abs/1502.00879v1

3. H. Nakajima, "Affine torus embeddings which are complete intersections". http://projecteuclid.org/euclid.tmj/1178228538

4. Lemma 2.7 in D. Dais, C. Haase, G. Zeigler, "All toric local complete intersection singularities admit projective crepant resolutions". http://projecteuclid.org/euclid.tmj/1178207533