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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 $\mathbb{P}(Q)$ 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 $\mathbb{P}(Q)$ 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.

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.

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

However, you cannot expect that they are 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 $\mathbb{P}(Q)$ 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.

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 $\mathbb{P}(Q)$ 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 $\mathbb{P}(Q)$ 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.

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

However, you cannot expect that they are 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. Analogously, $\mathbb{P}(1, \, 1, \,n)$ 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 $\mathbb{P}(Q)$ 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.

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

However, you cannot expect that they are 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 $\mathbb{P}(Q)$ 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.