About Affine Cones associated to ample line bundles

Here we will introduce some well-known results about the cones which is the simplest examples of terminal, canonical, etc. singularities. Here we will follow the book [Kol13]¹. We just consider $\mathrm{char}(k)=0$.

1. Basic Constructions

[Def 1.1] Let $X$ be a projective scheme with an ample line bundle $\scr{L}$.

(i) The affine cone over $X$ with conormal bundle $\scr{L}$ is

\[\mathsf{AC}(X,\mathscr{L}):=\mathrm{Spec}\bigoplus_{m\geq0}H^0(X,\mathscr{L}^{\otimes m}).\]

(ii) The projective cone over $X$ with conormal bundle $\scr{L}$ is

\[\mathsf{PC}(X,\mathscr{L}):=\mathrm{Proj}\bigoplus_{m\geq0}\left(\bigoplus_{r=0}^m H^0(X,\mathscr{L}^{\otimes r})x^{m-r}\right).\]

[Remark 1.2] (i) Note that we let $X\subset\mathbb{P}^n$, then we also have the classical affine cone in $\mathbb{A}^{n+1}$ and its closure $\mathsf{PC}(X)$ in $\mathbb{P}^{n+1}$ which is the classical projective cone (see also the more general case of classical cones in 13.37 in [GW1]²). Now consider $\mathsf{PC}(X,\mathscr{O}(1))\to\mathsf{PC}(X)$ is a natural finite morphism which is an isomorphism away from the vertex. If $X$ normal, then this is a normalization. So an advantage of our general notion is that if $X$ is normal (resp. S2) then $\mathsf{AC}(X,\mathscr{L})$ and $\mathsf{PC}(X,\mathscr{L})$ are also normal (resp. S2).

(ii) There are more general construction of cone of sheaves similar as ours, we refer 3.15 in [Kol13]¹. $\blacktriangleleft$

Here we mainly consider $\mathsf{AC}(X,\mathscr{L})$ and we have a natural partial resolution:

[Proposition 1.3] Let $X$ be a projective variety with an ample line bundle $\scr{L}$. Then we have

\[p_A:\mathsf{BAC}(X,\mathscr{L}):=\underline{\mathrm{Spec}}_X \bigoplus_{m\geq0}\mathscr{L}^{\otimes m}\to\mathsf{AC}(X,\mathscr{L})=\mathrm{Spec}\bigoplus_{m\geq0}H^0(X,\mathscr{L}^{\otimes m})\]

which is an isomorphism over the punctured affine cone $\mathsf{AC}(X,\mathscr{L})\backslash{\text{vertex}}$. The exceptional divisor is $E\cong X$ which is the zero section of $X$ in $\mathsf{BAC}(X,\mathscr{L})$.

[Remark 1.4] For projective case we also have the similar construction: we have

2. Properties

Now we will discuss some more results about them and we mainly focus on $\mathsf{AC}(X,\mathscr{L})$. Recall that we have a partial resolution:

[Proposition 2.1] Let $X$ be a projective scheme with an ample line bundle $\scr{L}$. Then

\[\mathbf{R}^ip_{A,*}\mathscr{O}_{\mathsf{BAC}(X,\mathscr{L})}\cong\bigoplus_{m\geq0}H^i(X,\mathscr{L}^{\otimes m}).\]

Proof. Now the exceptional divisor is $E$, we let its ideal sheaf is $\mathscr{I}\subset\mathscr{O}_{\mathsf{BAC}(X,\mathscr{L})}$. Actually by construction of $\mathsf{BAC}(X,\mathscr{L})$, we can easy to see that

\[\mathscr{O}_{\mathsf{BAC}(X,\mathscr{L})}/\mathscr{I}^m\cong\mathscr{O}_X\oplus\mathscr{L}\oplus\cdots\oplus\mathscr{L}^{\otimes m-1}.\]

Hence by Theorem on formal functions (c.f. Chapter 24 in [GW2]³)

[Proposition 2.2] Let $X$ be a normal projective variety with an ample line bundle $\scr{L}$. Then

(i) We have $\mathrm{Pic}(\mathsf{AC}(X,\mathscr{L}))=0$ and $\mathrm{Cl}(\mathsf{AC}(X,\mathscr{L}))=\mathrm{Cl}(X)/\mathbb{Z}\cdot\mathscr{L}$.

(ii) Let $\Delta_X$ be a $\mathbb{Q}$-divisor on $X$. By pull-back, we get $\mathbb{Q}$-divisors $\Delta_{\mathsf{AC}(X,\mathscr{L})}$ and $\Delta_{\mathsf{BAC}(X,\mathscr{L})}$. Assume that $K_X + \Delta_X$ is $\mathbb{Q}$-Cartier. Then we have

\[K_{\mathsf{BAC}(X,\mathscr{L})}+\Delta_{\mathsf{BAC}(X,\mathscr{L})}=\pi^*(K_X+\Delta_X)-E\]

where $\pi:\mathsf{BAC}(X,\mathscr{L})\to X$ is projection from the vertex and $E\cong X$ is the exceptional divisor of $p_A$.

Proof.

[Proposition 2.3] Let $X$ be a Cohen-Macaulay projective variety with an ample line bundle $\scr{L}$. Then $\mathsf{AC}(X,\mathscr{L})$ is Cohen-Macaulay if and only if $H^i(X,\mathscr{L}^{\otimes m})=0$ for any $m$ and any $0<i<\dim X$.

Proof.

[Proposition 2.4] Let $X$ be projective variety with rational singularities with an ample line bundle $\scr{L}$. Then the cone $\mathsf{AC}(X,\mathscr{L})$ has rational singularities if and only if $H^i(X,\mathscr{L}^{\otimes m})=0$ for any $m$ and any $0<i\leq\dim X$.

Proof.

[Example 2.5] Here we will introduce a log-canonical (lc) pair which is not a rational singularity and not Cohen-Macaulay using our theory.