In Prop. 4.1, p. 87 of the article "Ample vector bundles on curves" (Nagoya Math. J. 43 [1971], 73--89), R. Hartshorne states the following:

*Let $A$ be an abelian variety [over an alg. closed field $k$, say - my assumption]. Let $X$ be a non-singular subvariety of $A$.
Assume that every curve in $X$ generates $A$. Then the normal bundle $N$ to $X$ in $A$ is ample.*

The proof he provides is based on a criterion of Gieseker for ampleness and boils down to the following statement (X):

*Let $Y$ be a smooth curve over $k$ and let $\phi:Y\to A$ be a $k$-morphism, which is birational onto its image. Suppose that $\phi_*(Y)$ generates $A$.
Then the map $\phi^*:H^0(A,\Omega_{A/k})\to H^0(Y,\Omega_{Y/k})$ is injective.*

Hartshorne does not spell out a proof of (X) but it is likely that he had the following argument in mind:

"Proof": There is a factorisation $Y\to^{i}{\rm Jac}(Y)\to^{g} A$ of $\phi$, where $i$ is the morphism of $Y$ into the Jacobian ${\rm Jac}(Y)$ of $Y$ given by the choice of some $k$-point on $Y$ and $g$ exists because the Jacobian is an Albanese variety for $Y$. Now $i^*:H^0({\rm Jac}(Y),\Omega_{{\rm Jac}(Y)/k})\to H^0(Y,\Omega_{Y/k})$ is an isomorphism (this is a classical fact) and $g^*:H^0(A,\Omega_{A/k})\to H^0({\rm Jac}(Y),\Omega_{{\rm Jac}(Y)/k})$ is injective because the morphism $g$ is a surjective morphism between abelian varieties and is thus smooth.

(for this argument when $k={\bf C}$, see for instance Prop. 6.3.10 (p. 30) in R. Lazarsfeld, "Positivity in algebraic geometry. II." Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. No. 49. Springer-Verlag, Berlin, 2004.)

Now the "Proof" given above is apparently flawed, because when ${\rm char}(k)>0$, surjective morphisms between abelian varieties are not smooth in general. In fact, I think that (X) is false when ${\rm char}(k)>0$, although I do not have a counterexample handy. This is is also suggested by Serre's discussion of Albanese varieties in Exp. 10 of the "Séminaire Chevalley" (1958-1959) (see proof of Th. 3).

In fact it seems to me that Hartshorne's Prop. 4.1 is likely to be false when ${\rm char}(k)>0.$

My question is: is there a way to give a characteristic free proof of (X) ? Or are there classical counterexamples to (X) ? Is it well-known that Hartshorne's Prop. 4.1 is false when ${\rm char}(k)>0$ ?

I would also like to underline there is no implicit assumption on the characteristic of the base field in Hartshorne's paper. This is clear from the formulation of some corollaries of Prop. 4.1 (for instance Cor. 4.3).

Thank you in advance for your help.

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