Question about surface singularities

Question

Throughout, $X$ will be a projective surface. I am looking for examples of the following surface singularities,

I) A rational singularity that is not quotient. Obviously, it has to be non-Gorenstein, but I am trying to find such an example, more importantly, if there is some classification of such singularities.

II) A non-rational Du Bois singularity, and if there is some classification of such singularities.

Lastly, suppose we have the cone over an elliptic curve $C$, $X=\operatorname{Spec}(\oplus_{i\geq 0}H^0(C,\mathcal{O}(idP))$, where $P$ is a point on the curve. In the minimal resolution, the exceptional divisor $E$ will be the elliptic curve, but I was wondering what the self-intersection $E^2$ is. Also, is this a "simple elliptic singularity", in other words, is $h^0(X,\mathbb{R}^1f_*\mathcal{O}_{\tilde{X}})=1$, where $f:\tilde{X}\to X$ is the minimal resolution?

Answer 1

(1) As Marco Golla wrote in a comment: Némethi András, On the Ozsváth-Szabó invariant of negative definite plumbed 3-manifolds, Geom. Topol. 9 (2005), 991-1042 gives examples of rational non-quotient singularities.

(2) Ishii, Shihoko. "Du Bois singularities on a normal surface." Complex analytic singularities (1986): 153-163. Contains a discussion of elliptic Du Bois singularities on surfaces. There is also something akin to a classification of such singularities in this article. If I am not mistaken, Shihoko proves that any chain of smooth curves can appear as the reduction of the exceptional fiber of the minimal resolution of an isolated normal Du Bois singularity (Corollary 3.3).

(3) I think in general, if $Y \subset \mathbb{P}^n$ is a projectively normal embedding of degree $d$ and $X$ is the affine cone over $Y$ then the resolution $\tilde{X} \to X$ given by blowing up the cone point is the total space of the bundle $\mathcal{O}_Y(-1)$ with $Y$ embedded as the zero section. Therefore, the normal bundle of $Y \subset \tilde{X}$ is $\mathcal{O}_Y(-1)$. In particular, if $Y = C$ is a smooth curve then $C^2 = \deg \mathcal{O}_C(-1) = -d$.

In your case, $E$ is embedded via $\mathcal{O}_E(dP)$ which has degree $d$ so $E^2 = -d$.

To see why this claim is true, let $X = \mathrm{Spec} \left( \bigoplus_{n \ge 0} H^0(Y, \mathcal{O}_Y(n)) \right)$ be the affine cone and let $\pi : \tilde{X} = \mathbf{Spec}_Y \left( \bigoplus_{n \ge 0} \mathcal{O}_Y(n) \right) \to Y$ be the total space of the bundle $\mathcal{O}_Y(-1)$. Notice the dual in my convention! I do this so that the zero section is given by the ideal $$ \bigoplus_{n > 0} \mathcal{O}_Y(n) \subset \bigoplus_{n \ge 0} \mathcal{O}_Y(n) $$ and therefore the conormal bundle of the zero section is $\pi^* \mathcal{O}_Y(1)|_Y = \mathcal{O}_Y(1)$ showing the claim about the normal bundle.

Consider the map $\tilde{X} \to X$ induced by the pullback map on global sections: $$ H^0(Y, \mathcal{O}_Y(n)) \to H^0(\tilde{X}, \pi^* \mathcal{O}_Y(n)) \to H^0(\tilde{X}, \mathcal{O}_{\tilde{X}}) $$ using that $$ \pi_* \mathcal{O}_{\tilde{X}} = \bigoplus_{n \ge 0} \mathcal{O}_Y(n) $$

this gives a map of graded rings and hence a $\mathbb{G}_m$-equivariant map $\tilde{X} \to X$ which is the affinization, $\tilde{X} \to \mathrm{Spec}(H^0(\tilde{X}, \mathcal{O}_{\tilde{X}}))$. This is an isomorphism away from the zero section $Y \subset \tilde{X}$ because $\mathcal{O}_Y(1)$ is very ample. The maximal ideal $\mathfrak{m}$ generated by $H^0(Y, \mathcal{O}(1))$ cuts out the cone point (here we use that $Y \subset \mathbb{P}^n$ is projectively normal to ensure that the ring is generated in degree $1$). Since $\mathcal{O}_Y(1)$ is globally generated, $\mathfrak{m}$ generates the ideal sheaf of the zero section which is Cartier and hence $\tilde{X} \to X$ factors through the blowup $\tilde{X} \to \mathrm{Bl}_p X \to X$. To show that the first map is an isomorphism, use that the fiber over $p$ is exactly the tangent cone $C_p X \cong Y$ since $$ \mathrm{gr}_{\mathfrak{m}} = \bigoplus_{n \ge 0} \mathfrak{m}^n / \mathfrak{m}^{n+1} = \bigoplus_{n \ge 0} H^0(Y, \mathcal{O}_Y(n)) $$ again using generation in degree $1$. Therefore, $\tilde{X} \to \mathrm{Bl}_p X$ is a degree $1$ finite map of normal (because $X$ is normal by assumption) varities and hence an isomorphism.

If $Y \subset \mathbb{P}^n$ is not projectively normal, there are two things that one might call the ``affine cone'' that do not agree. We have $X$ as above and the closure in $\mathbb{A}^{n+1}$ of the preimage of $Y$ under $\mathbb{A}^{n-1} \setminus \{ 0 \} \to \mathbb{P}^n$ which is $\mathrm{Spec}(k[x_0, \dots, x_n]/I)$ where $I$ is the unique saturated ideal cutting out $Y$. If $Y$ is normal then $X$ is the normalization of this second cone which might be non-normal at the cone point.

(4) To compute $h^0(X, R^1 f_* \mathcal{O}_{\tilde{X}})$ we can use the theorem on formal functions. Since $R^1 f_* \mathcal{O}_{\tilde{X}}$ is supported at $p$, $$ h^0(X, R^1 f_* \mathcal{O}_{\tilde{X}}) = \dim_k (R^1 f_* \mathcal{O}_{\tilde{X}})_p = H^1(\hat{E}, \mathcal{O}_{\hat{E}}) $$ Where $\hat{E}$ is the formal fiber over $p$. Let $E_n$ be the $n^{\text{th}}$ neighborhood then there are exact sequences,

$$ 0 \to \mathcal{O}_E(-nE) \to \mathcal{O}_{E_{n+1}} \to \mathcal{O}_{E_n} \to 0 $$ Since $\mathcal{O}_E(-nE) = \mathcal{O}_E(n)$ because $\mathfrak{m}$ generates the ideal sheaf of $E$ and its conormal bundle is $\mathcal{O}_E(1) = \mathcal{O}_E(d P)$. For $n > 0$ this has no $H^1$ so $H^1(\mathcal{O}_{E_{n+1}}) = H^1(\mathcal{O}_{E_n})$ for $n > 0$. Then $H^1(\mathcal{O}_{E_1}) = k$ because $E$ is an elliptic curve so indeed, $h^0(X, R^1 f_* \mathcal{O}_{\tilde{X}}) = 1$. In general, this works for any curve embedded such that $\mathcal{O}_C(1)$ has vanishing $H^1$.

Answer 2

As already pointed out by @marco-golla, Némethi's article gives you an answer to your first question.

Regarding Du~Bois (DB) singularities: There are many interesting things about them:

1. You are probably asking for non-rational DB singularities, because you know that rational sings are DB. So, this is just for the record. But it is worth remembering that some people consider DB sings a generalization of rational sings.
2. A cone over an elliptic curve is indeed probably one of the simplest examples of a non-rtl DB sing. Well, perhaps the simplest would be a non-normal one, so probably the simplest is a node (curve) or a transversal intersection on a surface. Or, perhaps, for a more interesting non-normal example you can take the pinch point ($x^2=y^2z$). More generally, slc singularities, or their normal version, log canonical singularities are DB. There are plenty of examples of non-rational log canonical singularities (the cone over an elliptic curve is one).
3. In fact, it is quite easy to give an example of a non-rational DB sing: Take any smooth projective variety $Z$, embed it via the $d$-uple embedding, and let $X$ be the cone over that. If $d\gg 0$, then $X$ has DB sings. I don't know any published reference for this, so you might have to compute this yourself.
4. To see that the cone over an elliptic curve is DB, you can easily compute that it is log canonical: Let $X$ be a cone over an elliptic curve $C$. For simplicity let $C$ be embedded into $\mathbb P^2$, so it is a hypersurface and then so is $X$. In particular, $K_X$ is a Cartier divisor. Let the blow up of $X$ at the vertex be $\sigma:\widetilde X\to X$. Then $K_{\widetilde X}\sim \sigma^*K_X+aE$ for some $a\in\mathbb Z$, where $E$ is the exceptional divisor, which is isomorphic to $C$. Computing $0\sim K_E\sim (K_{\widetilde X}+E)|_E\sim \left(\sigma^*K_X+(a+1)E\right)|_E \sim (a+1)E|_E$ tells you that, since $E^2\neq 0$, $a=-1$. So, this is indeed a log canonical singularity and hence DB. In case you want to check out the slc/log canonical angle, the best source is Chapter 6 of Kollár's book Singularities of the Minimal Model Program. There is also a chapter on examples and a classification of various surface singularities.