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 $>n$.



Frobenius splitting of Fano varieties
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](https://arxiv.org/abs/0905.0404) 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](http://arxiv.org/pdf/1112.2425.pdf) 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.


