# Download Ample Subvarieties of Algebraic Varieties by Robin Hartshorne, C. Musili PDF

By Robin Hartshorne, C. Musili

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Xn ) is the maximal ideal of P . Consider the surjections P/ni+1 → P/ni , each deﬁned by an ideal of square zero. Starting with the map A → A/m2 ∼ = P/n2 , we can lift step by step to i get maps of A → P/n for each i, and hence a map g into the inverse limit, f g ˆ we have maps P → Aˆ → P with the property that which is P . Passing to A, g ◦ f is an isomorphism on P/n2 . It follows that g ◦ f is an automorphism of P (Ex. 1). Hence g ◦ f has no kernel, so f is injective. But f was surjective by construction, so f is an isomorphism, and Aˆ is regular.

In particular, by reason of dimension, the zero-scheme described in (b) is not in the closure of the component corresponding to sets of eight distinct points. It is therefore a nonsmoothable subscheme of P4 . (d) Show that the image of the natural map Hom(ΩR , B) → Hom(I/I 2 , B) has dimension 12, generated by the four homomorphisms that can be described as ∂/∂x, ∂/∂y, ∂/∂z, ∂/∂w, so that T 1 (B/k, B) has dimension 13. Thus this singularity is not rigid. Note. This is a slight variant of an example discovered by Iarrobino and Emsalem [72].

In Section 11 we show how an obstruction theory aﬀects the local ring of the corresponding parameter space, and in Section 12 we apply this to prove a classical bound on the dimension of the Hilbert scheme of curves in P3 . In Section 13 we describe one of Mumford’s examples of “pathologies” in algebraic geometry, a family of nonsingular curves in P3 whose Hilbert scheme is generically nonreduced. 6. Subschemes and Invertible Sheaves A general context in which to study higher-order deformations is the following.