Frobenius splitting of Fano varieties

Question

Dear MO,

Question 1. Do you know of an example of a Fano variety which is not Frobenius split?

Background

(1) A variety $X$ in characteristic $p$ is called Frobenius split if there is a "$p$-th root" map $\sigma: \mathcal{O}_X \to \mathcal{O}_X$, that is, an additive map satisfying $\sigma(f^p g) = f\sigma(g)$ and $\sigma(1) = 1$ (in particular, $\sigma(f^p) = f$, so that $\sigma$ is an $\mathcal{O}_X$-linear splitting of the Frobenius map $F: \mathcal{O}_X \to F_* \mathcal{O}_X$). Such varieties enjoy very nice properties, for example, $H^i(X, L)=0$ for $i>0$ for every ample line bundle $L$ on $X$.

In case $X$ is smooth and projective, $X$ is Frobenius split if and only if the map $F: H^{\dim X}(X, \omega_X) \to H^{\dim X}(X, F^* \omega_X)$ is nonzero. Note that $F^* \omega_X = \omega_X^p$. In particular, by Serre duality, $H^{\dim X}(X, \omega_X^p)^\vee = H^{0}(X, \omega_X^{1-p})$ is nonzero, that is, $(1-p)K_X$ is effective - so Frobenius split varieties are ,,on the Fano side''.

(2) A smooth projective variety $X$ is called Fano if $\omega_X^{-1}$ is ample. One can prove (Brion, Kumar Frobenius splitting methods in geometry and representation theory, Exercise 1.6.E5) that if $X$ is a Fano variety in characteristic $0$, then for $p\gg 0$ the reduction $X_p$ of $X$ mod $p$ is Fano and Frobenius split. This means that counterexamples to Question 1 might be difficult to find.

Further questions

Therefore, I am almost sure that, if counterexamples appear in the answers, they will have $\dim X$ (or other invariants of $X$, for example the degree of $K_X$ or its index) big compared to $p$. So I would like to ask:

Question 2. Can you find an effective bound $M = M(X) = M(\dim X, \ldots)$, depending on the dimension of $X$ and maybe other relatively simple invariants, such as the degree or index, such that whenever $X$ is a Fano variety in characteristic $p>M(X)$ then $X$ is Frobenius split. For example, does $M = 0$ (this is Question 1) or $M = n$ work?

Note. The $M = n$ case reminds me of the requirement in the theorem of Deligne-Illusie about decompositions of the de Rham complex that $p$ has to be $>n$.

Answer 1

For question 1, the following example works, and you'll see how to construct many more (in higher characteristic).

Set $k$ to be a perfect field of characteristic $2$.

$$X = \text{Proj } k[x,y,u,v]/\langle x^3+y^3+u^3+v^3 \rangle \subseteq \mathbb{P}^3_k.$$

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###Fano check:### I'm certain you already know this but... Certainly $K_X \sim (K_{\mathbb{P}^3} + X)|_X \sim (-4H + 3H)|_X = -H|_X$ where $H$ is the hyperplane on $\mathbb{P}^3_k$. Clearly also $X$ is smooth. Thus $X$ is Fano.

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###Frobenius splitting check:### This is slightly more involved. First we need a couple lemmas:

Lemma: [Probably due to Karen Smith] A projective variety $X$ is Frobenius split if and only if some/every section ring with respect to an ample divisor $$S_X = \bigoplus_{n \geq 0} O_X(nA).$$ is Frobenius split.

The proof is pretty easy. If $X$ is Frobenius split, so is $S_X$ (use the splitting on $S_X$ induced by that on $X$ on degrees divisible by $p$, throw the other degrees out). For the converse direction, it's not hard to show that if $S_X$ has a Frobenius splitting, it then has an appropriately graded Frobenius splitting (basically, use the fact that $Hom_{S_X}(F_* S_X, S_X)$ is generated by appropriately graded maps). It then induces maps on the associated sheaves. Care must be taken since $\widetilde{F_* S_X} \neq F_* \widetilde S_X = F_* O_X$, but the latter is a summand of the former and this is enough.

Now for the next lemma. If $J = \langle g_1, \dots, g_t \rangle$ is any ideal in a ring of characteristic $p$, we use $J^{[p]}$ to denote the ideal $\langle g_1^p, \dots, g_t^p \rangle$.

Lemma: [Fedder's Criterion] Suppose $p = \text{char} k$, then a ring $k[x_1, \dots, x_n]/I$ is Frobenius split at a point $\mathbb{m} \in \text{Spec} k[x_1, dots, x_n]$ if and only if $I^{[p]} : I \nsubseteq \mathbb{m}^{[p]}$.

See Lemma 1.6 in $F$-purity and rational singularity by Richard Fedder (1983, Transactions of the AMS). It's quite easy, but maybe more than I want to explain.

Corollary: If $p = 2$ and $I = \langle f \rangle$, then $R$ is Frobenius split at $\mathbb{m}$ if and only if $f \notin \mathbb{m}^{[2]}$. (in other characteristics, you have( $f^{p-1} \notin \mathbb{m}^{[p]}$).

Ok, now we verify that $X$ is not Frobenius split. Consider the section ring given our embedding, $S_X = k[x,y,u,v]/\langle x^3 + y^3 + u^3 + v^3 \rangle$. It is sufficient to show that $S_X$ is not Frobenius split, and that we can check at the origin since $S_X$ is a graded ring. Then we simply observe that $$ x^3+y^3+u^3+v^3 \in \langle x^2, y^2, u^2, v^2 \rangle. $$

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##Generalizations, question 2, and further references##

The result you mention about Fanos is actually generalized to log-Fano varieties in this paper: Globally $F$-regular and log Fano varieties due to Karen Smith and myself. Log Fano's are varieties where $-K_X$ is not necessarily ample, but where it is close. Some additional discussion of the lemmas above can also be found here.

Lots of people in commutative algebra have been exploring question of when various rings are $F$-split. For example, see THIS PAPER of Daniel Hernandez and references. I know there are people exploring exactly when graded hypersurfaces are $F$-split = $F$-pure but this is not public yet.

In terms of Question 2: effective bounds, I don't know any great ones off the top of my head. I'll try to get back to you on this.