1. Introduction
Let X and Y be irreducible smooth projective curves over an algebraically closed field k – there is no assumption on the characteristic of k – and let 
$f\,:\,X\,\longrightarrow \,Y$ be a generically smooth morphism. Then, we have 
${\mathcal O}_Y\, \subset\, f_*{\mathcal O}_X$. In [Reference Biswas and Parameswaran4] it was shown that the homomorphism of étale fundamental groups 
$f_*\, :\, \pi^{\rm et}_1(X)\, \longrightarrow\,\pi^{\rm et}_1(Y)$ induced by f is surjective if and only if 
${\mathcal O}_Y$ is the unique maximal semistable subsheaf of 
$f_*{\mathcal O}_X$. We call f to be genuinely ramified if 
${\mathcal O}_Y$ is the unique maximal semistable subsheaf of 
$f_*{\mathcal O}_X$. On the other hand, f is called primitive if the above homomorphism 
$f_*$ of étale fundamental groups is surjective [Reference Coskun, Larson and Vogt5]. So f is genuinely ramified if and only if it is primitive.
The main result of [Reference Biswas and Parameswaran4] says the following: If 
$f\,:\,X\,\longrightarrow \,Y$ is genuinely ramified, and E is a stable vector bundle on Y, then 
$f^*E$ is also stable. This was proved by investigating the quotient bundle 
$(f_*{\mathcal O}_X)/{\mathcal O}_Y$.
The dual vector bundle 
$((f_*{\mathcal O}_X)/{\mathcal O}_Y)^*$ is called the Tschirnhausen bundle for f (see [Reference Coskun, Larson and Vogt5]). The following is the main result of [Reference Coskun, Larson and Vogt5]: Let 
$f\,:\,X\,\longrightarrow \,Y$ be a general primitive degree r cover, where 
${\rm genus}(X)\,=\, g$ and 
${\rm genus}(Y)\,=\, h$, over an algebraically closed field of characteristic zero or greater than r. Then
(1)
$((f_*{\mathcal O}_X)/{\mathcal O}_Y)^*$ is semistable if $h\,=\, 1$, and(2)
$((f_*{\mathcal O}_X)/{\mathcal O}_Y)^*$ is stable if $h \,\geq \, 2$.
Note that the above mentioned result of [Reference Biswas and Parameswaran4] can be reformulated as follows: Let 
$f\,:\,X\,\longrightarrow \,Y$ be a generically smooth morphism between irreducible smooth projective curves. Then 
$f^*E$ is stable for every stable vector bundle E on Y if and only if:
\begin{equation*}
\mu_{\rm min}(((f_*{\mathcal O}_X)/{\mathcal O}_Y)^*) \, \gt \, 0.
\end{equation*}(Recall that µ min denotes the slope of the smallest quotient [Reference Huybrechts and Lehn9, p. 16, Definition 1.3.2].) See [Reference Coskun, Larson and Vogt5] for more on Tschirnhausen bundles.
A vector bundle on an irreducible smooth projective curve Z is called virtually globally generated if its pullback, under some surjective morphism to Z from some irreducible smooth projective curve, is generated by its global sections; see 
$\S$ 3.
We prove the following (see Theorem 3.3):
Let X and Y be irreducible smooth projective curves and:
\begin{equation*}
f\, :\, X\, \longrightarrow\, Y,
\end{equation*} a generically smooth morphism. Then 
$(f_*{\mathcal O}_X)^*$ is virtually globally generated.
Note that this implies that: 
$((f_*{\mathcal O}_X)/{\mathcal O}_Y)^*$ is virtually globally generated (see Corollary 3.5).
In Remark 3.6 it is shown that Corollary 3.5 fails in higher dimensions.
We prove the following (see Corollary 3.2):
Let 
$f\, :\, X\, \longrightarrow\, Y$ be a generically smooth morphism between two irreducible smooth projective curves. Then f is genuinely ramified if and only if 
$((f_*{\mathcal O}_X)/{\mathcal O}_Y)^*$ is ample.
It may be mentioned that the condition in Theorem 3.3 and Corollary 3.2 that f is generically smooth is essential. To give an example, take Y to be a smooth projective curve of genus at least two, and let 
$F_Y\, :\, Y\, \longrightarrow\, Y,$ be the absolute Frobenius morphism of Y. Then 
$(F_{Y*}{\mathcal O}_Y))/{\mathcal O}_Y$ is in fact ample.
2. Genuinely ramified maps, direct image and ampleness
The base field k is assumed to be algebraically closed. For a vector bundle E on an irreducible smooth projective curve X, if
\begin{equation*}
E_1\, \subset\, \cdots\, \subset\, E_{n-1}\, \subset\, E_n\,=\, E,
\end{equation*} is the Harder–Narasimhan filtration of E, then define 
$\mu_{\rm max}(E)\,:=\, \mu(E_1)$ and 
$\mu_{\rm min}(E)\, =\, \mu(E/E_{n-1})$ [Reference Huybrechts and Lehn9]. The subbundle 
$E_1\, \subseteq\, E$ is called the maximal semistable subsheaf of E.
Let X and Y be irreducible smooth projective curves and
\begin{equation}
f\, :\, X\, \longrightarrow\, Y,
\end{equation}a dominant generically smooth morphism. It is straight-forward to check that:
\begin{equation}
\mu_{\rm max}(f_*{\mathcal O}_X)\, =\, 0.
\end{equation} Indeed, 
$\mu_{\rm max}(f_*
{\mathcal O}_X)\, \leq\, 0$ because 
$\text{degree}({\mathcal O}_X)\,=\, 0$ [Reference Biswas and Parameswaran4, p. 12824, Lemma 2.2]. On the other hand, we have 
${\mathcal O}_Y\, \subset\, f_*{\mathcal O}_X$, which implies that 
$\mu_{\rm max}(f_*{\mathcal O}_X)\, \geq\,
0$, and thus (2.2) holds.
The following proposition was proved in [Reference Biswas and Parameswaran4].
Proposition 2.1. ([Reference Biswas and Parameswaran4, p. 12828, Proposition 2.6] and [Reference Biswas and Parameswaran4, p. 12830, Lemma 3.1])
The following five statements are equivalent:
(1) The maximal semistable subsheaf of
$f_*{\mathcal O}_X$ is ${\mathcal O}_Y$.(2)
$\dim H^0(X,\, f^*f_* {\mathcal O}_X)\,=\, 1$.(3) The fibre product
$X\times_Y X$ is connected.(4) The homomorphism of étale fundamental groups
$f_*\, :\, \pi^{\rm et}_1(X)\, \longrightarrow\,\pi^{\rm et}_1(Y)$ induced by f is surjective.(5) The map f does not factor through any nontrivial finite étale covering of Y.
Any morphism f as in (2.1) is called genuinely ramified if the (equivalent) statements in Proposition 2.1 hold [Reference Biswas and Parameswaran4, p. 12828, Definition 2.5].
Proposition 2.2. Let 
$f\, :\, X\, \longrightarrow\, Y$ be a genuinely ramified morphism of smooth projective curves. Then the vector bundle 
$((f_*{\mathcal O}_X)/{\mathcal O}_Y)^*$ is ample.
Proof. Since f is genuinely ramified, from Proposition 2.1 it follows that:
\begin{equation*}\mu_{\rm max}((f_*{\mathcal O}_X)/{\mathcal O}_Y)\, \lt \, 0,\end{equation*}and hence we have:
\begin{equation}
\mu_{\rm min}(((f_*{\mathcal O}_X)/{\mathcal O}_Y)^*)\,=\, - \mu_{\rm max}((f_*{\mathcal O}_X)/{\mathcal O}_Y)\, \gt \, 0.
\end{equation} When the characteristic of k is zero, a vector bundle W on Y is ample if and only if the degree of every nonzero quotient of W is positive [Reference Hartshorne7, p. 84, Theorem 2.4]. Therefore, from (2.3) we conclude that: 
$((f_*{\mathcal O}_X)/{\mathcal O}_Y)^*$ is ample, when the characteristic of k is zero. However, this characterization of ample bundles fails when the characteristic of k is positive (see [Reference Hartshorne7, Section 3] for such examples).
We will inductive construct a sequence of vector bundles 
$\{V_i\}_{i \geq 0}$ on Y. First set 
$V_0\, =\, {\mathcal O}_Y$. For any 
$i\,\geq\,1$, let 
$V_i\,=\, f_*f^*V_{i-1}$. Since we have
\begin{equation*}{\mathcal O}_Y\, \subset\, V_1\,=\, f_*f^*{\mathcal O}_Y\,=\, f_*{\mathcal O}_X,\end{equation*}it can be deduced that:
\begin{equation}
{\mathcal O}_Y\, \subset\, V_i,
\end{equation} for all 
$i\, \geq\, 0$. Indeed, this follows inductively, as the inclusion map 
${\mathcal O}_Y\, \hookrightarrow\, V_j$ produces:
\begin{equation*}
{\mathcal O}_Y \, \subset\, f_*{\mathcal O}_X \,=\, f_* f^*{\mathcal O}_Y \, \hookrightarrow\, f^*f_*V_j\,=\, V_{j+1}.
\end{equation*}This proves (2.4) inductively.
Next we will show that the subsheaf 
${\mathcal O}_Y$ in (2.4) is the maximal semistable subsheaf of Vi. This will also be proved using an inductive argument.
First, 
${\mathcal O}_Y$ is obviously the maximal semistable subsheaf of V 0. Next, from Proposition 2.1 we know that 
${\mathcal O}_Y$ is the maximal semistable subsheaf of V 1 (recall that f is genuinely ramified). Let
\begin{equation*}
{\mathcal O}_Y \,=\, E^1_1\, \subset\, E^1_2\, \subset\, \cdots\, \subset\, E^1_{n_1-1}\, \subset\, E^1_{n_1}\,=\, V_1,
\end{equation*} be the Harder–Narasimhan filtration of V 1. Since 
$f^*W$ is semistable if W is so (see [Reference Biswas and Parameswaran4, pp. 12823–12824, Remark 2.1]), we conclude that:
\begin{equation}
{\mathcal O}_X \,=\, f^*E^1_1\, \subset\, \cdots\, \subset\, f^*E^1_{n_1-1}\, \subset\, f^*E^1_{n_1}\,=\, f^*V_1,
\end{equation} is the Harder–Narasimhan filtration of 
$f^*V_1$.
For any vector bundle B on X, we have 
$\mu_{\rm max}(f_* B) \, \leq\, \mu_{\rm max}(B)/\text{degree}(f)$ [Reference Biswas and Parameswaran4, Lemma 2.2, p. 12824]. In view of the Harder–Narasimhan filtration in (2.5), this implies that:
\begin{equation*}
\mu_{\rm max}((f_*f^*E^1_{j+1})/(f_*f^*E^1_{j})) \, \lt \, 0,
\end{equation*} for all 
$1\,\leq\, j\, \, \leq\, n_1-1$, because 
$\mu_{\rm max}((f^*E^1_{j+1})/(f^*E^1_{j})) \, \lt \, 0$. Also, as noted before, the maximal semistable subsheaf of 
$f_*{\mathcal O}_X$ is 
${\mathcal O}_Y$. Combining these we conclude that 
${\mathcal O}_Y$ is the maximal semistable subsheaf of 
$f_*f^*V_1\,=\, V_2$.
The above argument works inductively. To explain this, let
\begin{equation*}
{\mathcal O}_Y \,=\, E^\ell_1\, \subset\, E^\ell_2\, \subset\, \cdots\, \subset\, E^\ell_{n_\ell-1}\, \subset\, E^\ell_{n_\ell}
\,=\, V_\ell,
\end{equation*} be the Harder–Narasimhan filtration of 
$V_\ell$. As before, we have:
\begin{equation*}
\mu_{\rm max}((f_*f^*E^\ell_{j+1})/(f_*f^*E^\ell_{j})) \, \lt \, 0,
\end{equation*} for all 
$1\,\leq\, j\, \, \leq\, n_\ell-1$, because 
$\mu_{\rm max}((f^*E^\ell_{j+1})/(f^*E^\ell_{j})) \, \lt \, 0$. Using this together with the fact that the maximal semistable subsheaf of 
$f_*{\mathcal O}_X$ is 
${\mathcal O}_Y$ we conclude that 
${\mathcal O}_Y$ is the maximal semistable subsheaf of 
$f_*f^*V_\ell\,=\, V_{\ell+1}$.
The projection formula (see [Reference Hartshorne8, p. 124, Ch. II, Ex. 5.1(d)], [Reference Serre11]) gives that 
$V_{i+1} \,=\, f_*f^*V_i\,=\,
V_i\otimes (f_*{\mathcal O}_X)$ for all 
$i\, \geq\, 1$. This implies that:
\begin{equation}
V_i\,=\, (f_*{\mathcal O}_X)^{\otimes i}\,=\, V^{\otimes i}_1,
\end{equation} for all 
$i\, \geq\, 1$.
Now we assume that the characteristic of k is positive (recall that the proposition was proved when the characteristic of k is zero). Let p be the characteristic of k. Let
\begin{equation*}F_Y\, :\, Y\, \longrightarrow\, Y\end{equation*}be the absolute Frobenius morphism of Y. For any vector bundle W on Y, we have the inclusion:
\begin{equation*}
F^*_Y W\, \subset\, W^{\otimes p},
\end{equation*} it is constructed using the map 
$W\, \longrightarrow\, W^{\otimes p}$ defined by 
$v\, \longmapsto\, v^{\otimes p}$. Therefore, from (2.6) we have
\begin{equation}
(F^n_Y)^* V_1\, \subset\, (V_1)^{\otimes np}\,=\, V_{np}
\end{equation} for all 
$n\, \geq\, 1$. Since 
${\mathcal O}_Y$ in (2.4) is the maximal semistable subsheaf of Vi, from (2.7) we have:
\begin{equation*}
(F^n_Y)^* (V_1/{\mathcal O}_Y)\,=\, ((F^n_Y)^* V_1)/{\mathcal O}_Y
\, \subset\, V_{np}/{\mathcal O}_Y,
\end{equation*}and
\begin{equation}
\mu_{\rm max}((F^n_Y)^* (V_1/{\mathcal O}_Y)) \, \lt \, 0,
\end{equation} because 
$\mu_{\rm max}(V_{np}/{\mathcal O}_Y) \, \lt \, 0$.
From (2.8) it follows that:
\begin{equation*}
\mu_{\rm min}((F^n_Y)^* (V_1/{\mathcal O}_Y)^*) \,=\, - \mu_{\rm max}((F^n_Y)^* (V_1/{\mathcal O}_Y)) \, \gt \, 0,
\end{equation*} for all 
$n\, \geq\, 1$. This implies that 
$(V_1/{\mathcal O}_Y)^*\,=\, ((f_*{\mathcal O}_X)/{\mathcal O}_Y)^*$ is ample [Reference Biswas1, p. 542, Theorem 2.2].
3. Virtual global generation
Let E be a vector bundle on an irreducible smooth projective curve Z. It will be called virtually globally generated if there is a finite surjective morphism:
\begin{equation*}
\phi\, :\,M\, \longrightarrow\, Z,
\end{equation*} from an irreducible smooth projective curve M such that 
$\phi^*E$ is generated by its global sections. The vector bundle E is called étale trivializable if there is a pair 
$(M,\, \phi)$ as above such that ϕ is étale and 
$\phi^*E$ is trivializable.
If 
$\text{degree}(E)\, \lt \, 0$, then E is not virtually globally generated. More generally, E is not virtually globally generated if it admits a quotient of negative degree. To give a nontrivial example of vector bundle which is not virtually globally generated, let Z be a compact connected Riemann surface of genus g, with 
$g\, \geq\, 2$. Note that the free group of g generators is a quotient of 
$\pi_1(Z)$. To see this, express 
$\pi_1(Z)$ as the quotient of the free group, with generators 
$a_1,\, \cdots,\, a_g,\, b_1,\,
\cdots,\, b_g$, by the single relation 
$\prod_{i=1}^g [a_i,\, b_i]\,=\, 1$. Then the quotient of 
$\pi_1(Z)$ by the normal subgroup generated by 
$b_1,\, \cdots,\, b_g$ is the free group generated by 
$a_1,\, \cdots,\, a_g$. Therefore, there is homomorphism:
\begin{equation*}
\rho\, \, :\,\, \pi_1(Z)\,\, \longrightarrow\,\, \text{U}(r),
\end{equation*} where 
$\text{U}(r)$ is the group of r × r unitary matrices, such that 
$\rho(\pi_1(Z))$ is a dense subgroup of 
$\text{U}(r)$ (the subgroup of 
$\text{U}(r)$ generated by two general elements of it is dense in 
${\rm U}(r)$). Let E denote the flat unitary vector bundle on Z given by ρ. This vector bundle E is stable of degree zero [Reference Narasimhan and Seshadri10]. Let M be a compact connected Riemann surface and
\begin{equation*}
\phi\, :\,M\, \longrightarrow\, Z,
\end{equation*}a surjective holomorphic map. Since the image of the induced homomorphism:
\begin{equation*}
\phi_*\,\, :\,\,\pi_1(M)\,\, \longrightarrow\,\, \pi_1(Z),
\end{equation*} is a subgroup of 
$\pi_1(Z)$ of finite index, the image of the following composition of homomorphisms:
\begin{equation*}
\pi_1(M)\, \,\stackrel{\phi_*}{\longrightarrow}\, \,\pi_1(Z) \,\, \stackrel{\rho}{\longrightarrow}\,\, \text{U}(r),
\end{equation*} is a dense subgroup of 
$\text{U}(r)$. This implies that 
$\phi^*E$ is a stable vector bundle of degree zero [Reference Narasimhan and Seshadri10]. In particular, we have
\begin{equation*}
H^0(M,\, \phi^*E)\,=\, 0.
\end{equation*}Hence E is not virtually globally generated.
Theorem 3.1. Let X and Y be irreducible smooth projective curves over k and
\begin{equation*}
f\, :\, X\, \longrightarrow\, Y,
\end{equation*} a generically smooth morphism. Then 
$f_*{\mathcal O}_X$ fits in a short exact sequence of vector bundles on Y:
\begin{equation*}
0\, \longrightarrow\, E \, \longrightarrow\, f_*{\mathcal O}_X \,
\longrightarrow\, V \, \longrightarrow\, 0,
\end{equation*} where E is étale trivializable and 
$V^*$ is ample.
Proof. Let
\begin{equation}
S^f\, \subset\,f_*{\mathcal O}_X,
\end{equation} be the maximal semistable subbundle. From (2.2) we know that 
$\text{degree}(S^f)\,=\, 0$.
The algebra structure of 
${\mathcal O}_X$ produces an algebra structure on the direct image 
$f_*{\mathcal O}_X$. The subsheaf Sf in (3.1) is a subalgebra. Moreover, there is an étale covering 
$g\, :\, Z\, \longrightarrow\, Y$ such that:
• f factors through g, meaning there is a morphism:
(3.2)\begin{equation}
h\, :\, X\, \longrightarrow\, Z
\end{equation}such that
$g\circ h\,=\, f$, and• the subsheaf
$g_*{\mathcal O}_Z \, \subset\,f_*{\mathcal O}_X$ coincides with Sf.
(See the proof of [Reference Biswas and Parameswaran4, p. 12828, Proposition 2.6] and [Reference Biswas and Parameswaran4, p. 12829, (2.13)].) Moreover, the map h in (3.2) is genuinely ramified [Reference Biswas and Parameswaran4, p. 12829, Corollary 2.7].
Consider the short exact sequence of vector bundles on Y:
\begin{equation}
0\, \longrightarrow\, S^f \, \longrightarrow\, f_*{\mathcal O}_X \, \longrightarrow\, Q\,:=\,(f_*{\mathcal O}_X)/S^f
\, \longrightarrow\, 0.
\end{equation} The pullback 
$g^*Q$, where Q is the vector bundle in (3.3), is identified with 
$(h_*{\mathcal O}_X)/{\mathcal O}_Z$, where h is the map in (3.2). From Proposition 2.2 we know that 
$((h_*{\mathcal O}_X)/{\mathcal O}_Z)^*$ is ample, Since 
$((h_*{\mathcal O}_X)/{\mathcal O}_Z)^*\,=\, g^*Q^*$, this implies that 
$Q^*$ in (3.3) is ample (see [Reference Hartshorne6, p. 73, Proposition 4.3]).
Since 
$Q^*$ is ample, in view of (3.3), it suffices to prove that Sf is a finite vector bundle.
Fix an étale Galois covering 
$\varphi\, :\, M\, \longrightarrow\, Y$ that dominates g. In other words, there is a morphism:
\begin{equation*}
\beta \, :\, M\, \longrightarrow\, Z
\end{equation*} such that 
$g\circ\beta\,=\, \varphi$. Since φ is an étale Galois covering, the vector bundle 
$\varphi^*\varphi_* {\mathcal O}_M$ is trivializable. On the other hand,
\begin{equation*}
S^f\,=\, g_*{\mathcal O}_Z\, \subset\, \varphi_*{\mathcal O}_M,
\end{equation*} and Sf is a subbundle of 
$\varphi_*{\mathcal O}_M$. Consider the subbundle
\begin{equation}
\varphi^*S^f\, \subset\, \varphi^*\varphi_* {\mathcal O}_M.
\end{equation} We have 
$\text{degree}(\varphi^*S^f)\,=\, 0$, because 
$\text{degree}(S^f)\,=\, 0$, and we also know that 
$\varphi^*\varphi_* {\mathcal O}_M$ is trivializable. Consequently, the subbundle 
$\varphi^*S^f$ in (3.4) is also trivializable. Hence Sf is étale trivializable.
Corollary 3.2. Let 
$f\, :\, X\, \longrightarrow\, Y$ be a generically smooth morphism between two irreducible smooth projective curves. Then f is genuinely ramified if and only if 
$((f_*{\mathcal O}_X)/{\mathcal O}_Y)^*$ is ample.
Proof. In view of Proposition 2.2 it suffices to show that 
$((f_*{\mathcal O}_X)/
{\mathcal O}_Y)^*$ is not ample if f is not genuinely ramified. If f is not genuinely ramified, then 
$\text{rank}(S^f)\, \geq\, 2$ (see (3.1)). Hence 
$(S^f/{\mathcal O}_Y)^*$ is a quotient of 
$((f_*{\mathcal O}_X)/{\mathcal O}_Y)^*$ (see (3.3)). But 
$\text{degree}((S^f/{\mathcal O}_Y)^*)\,=\, 0$ because 
$\text{degree}((S^f)\,=\, 0$. Now 
$((f_*{\mathcal O}_X)/
{\mathcal O}_Y)^*$ is not ample because its quotient 
$(S^f/{\mathcal O}_Y)^*$ is not ample.
Theorem 3.3. Let X and Y be irreducible smooth projective curves and
\begin{equation*}
f\, :\, X\, \longrightarrow\, Y,
\end{equation*} a generically smooth morphism. Then 
$(f_*{\mathcal O}_X)^*$ is virtually globally generated.
Proof. First assume that the characteristic of k is zero. We will show that the short exact sequence in (3.3) splits. First, the inclusion map 
${\mathcal O}_Z \, \hookrightarrow\, h_*{\mathcal O}_X$ splits naturally, where h is the map in (3.2); in other words,
\begin{equation*}
h_*{\mathcal O}_X\,=\, {\mathcal O}_Z\oplus F\, ;
\end{equation*} the fibre of F over any 
$z\, \in\, Z$ is the space of functions on 
$h^{-1}(z)$ whose sum is zero. Now we have
\begin{equation}
f_*{\mathcal O}_X\,=\, g_*h_*{\mathcal O}_X\,=\, g_*({\mathcal O}_Z\oplus F)\,=\,
(g_*{\mathcal O}_Z)\oplus g_* F\,=\, S^f\oplus g_*F.
\end{equation} From (3.3) and (3.5) it follows that the vector bundle 
$g_*F$ is isomorphic to Q. Therefore, from (3.5) we have
\begin{equation}
(f_*{\mathcal O}_X)^*\,=\, (S^f)^*\oplus Q^*.
\end{equation} Now 
$(S^f)^*$ is virtually globally generated because Sf is étale trivializable, and 
$Q^*$ is virtually globally generated because 
$Q^*$ is ample by Theorem 3.1 (see [Reference Biswas and Parameswaran3, p. 46, Theorem 3.6]). Therefore, from (3.6) it follows that 
$(f_*{\mathcal O}_X)^*$ is virtually globally generated.
Next assume that the characteristic of k is positive. As before,
\begin{equation*}F_Y\, :\, Y\, \longrightarrow\, Y\end{equation*} is the absolute Frobenius morphism of Y. Consider the exact sequence in (3.3); recall that Sf is the maximal semistable subsheaf of 
$f_*{\mathcal O}_X$. Therefore, there is an integer n 0 such that for all 
$n\, \geq\, n_0$, we have
\begin{equation*}
(F^n_Y)^*f_*{\mathcal O}_X\,=\, (F^n_Y)^* S^f\oplus (F^n_Y)^* Q
\end{equation*}[Reference Biswas and Parameswaran2, p. 356, Proposition 2.1]. Therefore,
\begin{equation}
(F^n_Y)^*(f_*{\mathcal O}_X)^*\,=\, (F^n_Y)^* (S^f)^*\oplus (F^n_Y)^* Q^*.
\end{equation} Now 
$(F^n_Y)^* (S^f)^*$ is virtually globally generated because Sf is étale trivializable and the Frobenius morphism commutes with étale morphisms. Also, 
$Q^*$ is virtually globally generated because 
$Q^*$ is ample by Theorem 3.1 (see [Reference Biswas and Parameswaran2, p. 357, Theorem 2.2]). Therefore, from (3.7) it follows that 
$(f_*{\mathcal O}_X)^*$ is virtually globally generated.
Corollary 3.4. Let X and Y be irreducible smooth projective curves and
\begin{equation*}
f\, :\, X\, \longrightarrow\, Y,
\end{equation*}a generically smooth morphism. Then the following statements hold:
• If the characteristic of k is zero, then
\begin{equation*}
(f_*{\mathcal O}_X)^*\,=\, E\oplus A,
\end{equation*}where E is étale trivializable and A is ample.
• If the characteristic of k is positive, then there is an integer n such that:
\begin{equation*}
(F^n_Y)^*(f_*{\mathcal O}_X)^*\,=\, E\oplus A,
\end{equation*}where E is étale trivializable and A is ample.
Proof. In view of Theorem 3.3, this follows immediately from [Reference Biswas and Parameswaran3, p. 40, Theorem 1.1].
Corollary 3.5. Let X and Y be irreducible smooth projective curves and
\begin{equation*}
f\, :\, X\, \longrightarrow\, Y,
\end{equation*} a generically smooth morphism. Then 
$((f_*{\mathcal O}_X)/{\mathcal O}_Y)^*$ is virtually globally generated.
Proof. From Theorem 3.3 we know that there is a finite surjective map:
\begin{equation*}
\phi\, :\,M\, \longrightarrow\, Y,
\end{equation*} such that 
$\phi^*(f_*{\mathcal O}_X)^*$ is generated by its global sections. We have the short exact sequence of vector bundles on M:
\begin{equation}
0\, \longrightarrow\, \phi^*((f_*{\mathcal O}_X)/{\mathcal O}_Y)^* \, \longrightarrow\,
\phi^*(f_*{\mathcal O}_X)^* \, \longrightarrow\, \phi^*({\mathcal O}_Y)^*\,=\,
{\mathcal O}_M \, \longrightarrow\, 0.
\end{equation} Since 
$\phi^*(f_*{\mathcal O}_X)^*$ is generated by its global sections, it has a section that projects to a nonzero section of 
${\mathcal O}_M$. Choosing such a section we obtain a splitting of (3.8). Since 
$\phi^*(f_*{\mathcal O}_X)^*$ is generated by its global sections, its direct summand 
$\phi^*((f_*{\mathcal O}_X)/{\mathcal O}_Y)^*$ is also generated by its global sections.
Remark 3.6. Corollary 3.5 is not valid in higher dimensions. To give an example, let X denote 
${\mathbb C}{\mathbb P}^2$ blown up at the point 
$(1,\, 0,\, 0)$. The involution of 
${\mathbb C}{\mathbb P}^2$ defined by 
$(x,\, y,\, z)\, \longmapsto\, (x,\,-y,\, -z)$ lifts to X; let
\begin{equation*}
\tau\, :\, X\, \longrightarrow\, X,
\end{equation*} be this lifted involution. Set 
$Y\,:=\, X/({\mathbb Z}/2{\mathbb Z})$ to be the quotient of X for the action of 
${\mathbb Z}/2{\mathbb Z}$ given by τ. Let
\begin{equation*}
f\, :\, X\, \longrightarrow\, X/({\mathbb Z}/2{\mathbb Z})\,=\, Y,
\end{equation*} be the quotient map. Then the line bundle 
$((f_*{\mathcal O}_X)/{\mathcal O}_Y)^*$ is not virtually globally generated. To see this, first note that the line bundle 
$f^*(((f_*{\mathcal O}_X)/{\mathcal O}_Y)^*)$ is virtually globally generated if 
$((f_*{\mathcal O}_X)/{\mathcal O}_Y)^*$ is virtually globally generated. But
\begin{equation*}f^*(((f_*{\mathcal O}_X)/{\mathcal O}_Y)^*)\, =\, {\mathcal O}_{X}(D_e+D_\infty),\end{equation*} where 
$D_e\, \subset\, X$ is the exceptional divisor and 
$D_\infty\, \subset\, X$ is the inverse image of
\begin{equation*}
\{(0,\, y,\, z)\,\in\, {\mathbb C}{\mathbb P}^2\,\, \mid\,\, y,\, z\, \in\, {\mathbb C}\}\, \subset\,
{\mathbb C}{\mathbb P}^2.
\end{equation*} It is easy to see that 
${\mathcal O}_{X}(D_e+D_\infty)$ is not virtually globally generated. Indeed, if
\begin{equation*}\varpi\, :\, Z\, \longrightarrow\, X\end{equation*} is a finite surjective proper map, then every section of 
$\varpi^*{\mathcal O}_{X}(D_e+D_\infty)$ vanishes on 
$\varpi^{-1}(D_e)$.
Funding statement
The first author is partially supported by a J. C. Bose Fellowship (JBR/2023/000003).
Competing interests
None declared.
