---
title: Virtual Global Generation (VGG)
url: https://www.emergentmind.com/topics/virtual-global-generation-vgg
type: topic
---

# Virtual Global Generation (VGG)

Virtual global generation (VGG) is a relaxation of global generation for vector bundles that is formulated by allowing finite pullback. On a smooth projective curve \(X\), a vector bundle \(E\) is virtually globally generated if there exists a nonconstant morphism \(f:Y\to X\) from a smooth irreducible projective curve \(Y\) such that \(f^*E\) is globally generated [2509.25653]. In the curve case this notion is tightly controlled by slope-theoretic positivity and finite or étale-trivializable factors, while in higher dimensions several natural extensions cease to agree [2509.25653]. A particularly important curve-level application is the Tschirnhausen bundle
\[
E:=\big((f_*\mathcal O_X)/\mathcal O_Y\big)^*
\]
attached to a generically smooth morphism \(f:X\to Y\) of irreducible smooth projective curves: it is virtually globally generated, and it is ample exactly when \(f\) is genuinely ramified [2403.15231].

## 1. Definitions on curves and the basic geometric setup

Let \(Y\) be a smooth projective curve over an algebraically closed field \(k\), and let \(E\) be a vector bundle on \(Y\). Global generation means that the natural evaluation map
\[
ev:H^0(Y,E)\otimes \mathcal O_Y\to E
\]
is surjective [2403.15231]. Following Section 3 of [2403.15231], \(E\) is virtually globally generated if there exists a finite surjective morphism \(g:M\to Z\) from an irreducible smooth projective curve \(M\) such that \(g^*E\) is generated by its global sections. The same paper also uses the related notion “étale trivializable”: \(E\) is étale trivializable if there exists an étale cover \(g:M\to Z\) with \(g^*E\) trivial [2403.15231].

The curve definition used in [2509.25653] is equivalent in spirit: if \(X\) is a smooth projective curve and \(E\) a vector bundle on \(X\), then \(E\) is VGG if there exists a nonconstant morphism \(f:Y\to X\) from a smooth irreducible projective curve \(Y\) such that \(f^*E\) is globally generated [2509.25653]. The paper further states that on curves this is equivalent to several classical characterizations, and quotes the fundamental criterion from [BP2, Theorem 1.1]: if \(\operatorname{char}(k)=0\), then \(E\) is VGG if and only if \(E\) is a direct sum of a finite vector bundle and an ample vector bundle; if \(\operatorname{char}(k)=p>0\), then \(E\) is VGG if and only if \(F^{n*}E\) is a sum of a finite vector bundle and an ample vector bundle for some \(n\ge 1\), where \(F:X\to X\) is the absolute Frobenius [2509.25653].

A standard geometric source of VGG in [2403.15231] starts with a generically smooth morphism
\[
f:X\to Y
\]
between irreducible smooth projective curves over an algebraically closed field of arbitrary characteristic. Here “generically smooth” means that \(f\) is smooth over a dense open subset of \(Y\); for morphisms of curves this is equivalent to \(f\) being separable on the generic point [2403.15231]. Since \(X\) and \(Y\) are projective curves and \(f\) is nonconstant, \(f\) is finite, and because the source and target are smooth curves, \(f_*\mathcal O_X\) is a locally free \(\mathcal O_Y\)-sheaf [2403.15231].

## 2. Pushforward algebras and the Tschirnhausen bundle

The inclusion \(\mathcal O_Y\subset f_*\mathcal O_X\) induced by the unit of the \(\mathcal O_Y\)-algebra structure yields a short exact sequence
\[
0\to \mathcal O_Y\to f_*\mathcal O_X\to Q\to 0,
\]
where
\[
Q:=(f_*\mathcal O_X)/\mathcal O_Y
\]
[2403.15231]. The dual bundle
\[
E:=Q^*=\big((f_*\mathcal O_X)/\mathcal O_Y\big)^*
\]
is the Tschirnhausen bundle associated to \(f\) [2403.15231].

If \(f\) is finite flat of degree \(r\), then
\[
\operatorname{rank}(f_*\mathcal O_X)=r,\qquad \operatorname{rank}(Q)=r-1,\qquad \operatorname{rank}(E)=r-1
\]
[2403.15231]. The bundle \(f_*\mathcal O_X\) carries a natural \(\mathcal O_Y\)-algebra structure, with \(\mathcal O_Y\) as the unit subalgebra [2403.15231]. This exact sequence is the starting point for both the ampleness criterion and the virtual global generation theorem.

A structural refinement is given by Theorem 3.1 of [2403.15231]: there is a short exact sequence
\[
0\to E_0\to f_*\mathcal O_X\to V\to 0,
\]
where \(E_0\) is étale trivializable and \(V^*\) is ample. Here \(E_0\) is identified with the maximal semistable subbundle \(S_f\subset f_*\mathcal O_X\); it has degree \(0\) and arises from an étale factorization \(g:Z\to Y\) and \(h:X\to Z\) such that \(f=g\circ h\) and \(S_f=g_*\mathcal O_Z\) [2403.15231]. The quotient is then \(V=(f_*\mathcal O_X)/S_f\), and the dual positivity of this quotient is inherited from the genuinely ramified part of the factorization [2403.15231].

This decomposition is the mechanism behind VGG in the curve case. In characteristic \(0\), the factorization \(f=g\circ h\) with \(h\) genuinely ramified gives
\[
h_*\mathcal O_X=\mathcal O_Z\oplus F,
\]
where \(F\) is the “trace-zero” summand, and after applying \(g_*\) one gets
\[
f_*\mathcal O_X=g_*\mathcal O_Z\oplus g_*F=S_f\oplus Q,
\qquad
(f_*\mathcal O_X)^*=S_f^*\oplus Q^*
\]
[2403.15231]. Since \(S_f\) is étale trivializable and \(Q^*\) is ample, both summands are VGG, hence \((f_*\mathcal O_X)^*\) is VGG; because \(Q^*\) is a direct summand, \(E\) is also VGG [2403.15231].

## 3. Genuine ramification, slopes, and ampleness

The notion of genuine ramification is central in [2403.15231]. A dominant generically smooth morphism \(f:X\to Y\) of smooth projective curves is genuinely ramified if \(\mathcal O_Y\) is the unique maximal semistable subsheaf of \(f_*\mathcal O_X\) [2403.15231]. The paper records equivalent conditions: the maximal semistable subsheaf of \(f_*\mathcal O_X\) is \(\mathcal O_Y\); \(\dim H^0(X,f^*f_*\mathcal O_X)=1\); the fiber product \(X\times_Y X\) is connected; the induced homomorphism
\[
f_*:\pi_1^{\mathrm{\acute et}}(X)\to \pi_1^{\mathrm{\acute et}}(Y)
\]
is surjective; and \(f\) does not factor through any nontrivial finite étale covering of \(Y\) [2403.15231]. In the terminology of [CLV], such an \(f\) is “primitive,” and the paper notes that “genuinely ramified” is equivalent to “primitive” [2403.15231].

The principal ampleness statement is Corollary 3.2: for a generically smooth morphism \(f:X\to Y\) between irreducible smooth projective curves,
\[
f \text{ is genuinely ramified } \Longleftrightarrow E=\big((f_*\mathcal O_X)/\mathcal O_Y\big)^* \text{ is ample}
\]
[2403.15231]. The slope-theoretic input begins with the identity
\[
\mu_{\max}(f_*\mathcal O_X)=0
\]
(Equation (2.2)), because \(\mathcal O_Y\subset f_*\mathcal O_X\) gives a semistable subbundle of slope \(0\), while general bounds imply \(\mu_{\max}(f_*\mathcal O_X)\le 0\) [2403.15231].

If \(f\) is genuinely ramified, then the uniqueness of \(\mathcal O_Y\) as maximal semistable subbundle forces
\[
\mu_{\max}(Q)<0,
\qquad
\mu_{\min}(E)=-\mu_{\max}(Q)>0
\]
[2403.15231]. In characteristic \(0\), ampleness of a vector bundle \(W\) on a curve is equivalent to the property that every nonzero quotient line bundle of \(W\) has positive degree, and \(\mu_{\min}(W)>0\) implies ampleness [2403.15231]. In characteristic \(p>0\), the paper uses an inductive construction and Frobenius pullbacks together with [Bi, Theorem 2.2] to conclude ampleness from positivity of minimal slopes after Frobenius [2403.15231].

Conversely, if \(f\) is not genuinely ramified, then the maximal semistable subbundle \(S_f\subset f_*\mathcal O_X\) has degree \(0\) and rank at least \(2\), and \((S_f/\mathcal O_Y)^*\) is a quotient of \(E\) of degree \(0\) [2403.15231]. A vector bundle on a curve with a quotient of degree \(0\) cannot be ample, so \(E\) is not ample [2403.15231]. This directly separates ampleness from VGG: the Tschirnhausen bundle is always VGG in the curve setting of the theorem, but it is ample only in the genuinely ramified case [2403.15231].

## 4. The VGG theorem for curves and Frobenius methods

The main VGG statement in [2403.15231] is Theorem 3.3: if \(f:X\to Y\) is a generically smooth morphism between irreducible smooth projective curves, then \((f_*\mathcal O_X)^*\) is virtually globally generated. Corollary 3.5 then gives the corresponding statement for the Tschirnhausen bundle:
\[
E=\big((f_*\mathcal O_X)/\mathcal O_Y\big)^*
\]
is virtually globally generated [2403.15231].

In characteristic \(0\), the proof combines the decomposition
\[
(f_*\mathcal O_X)^*=S_f^*\oplus Q^*
\]
with two inputs: \(S_f\) is étale trivializable, hence \(S_f^*\) is VGG, and \(Q^*\) is ample, while ample bundles on curves are VGG by [BP2, Theorem 3.6] [2403.15231]. The same decomposition also shows that \(E=Q^*\) is VGG, since it is a direct summand [2403.15231].

In positive characteristic, the argument passes to Frobenius pullback. Let \(F_Y\) be the absolute Frobenius. For \(n\gg 0\),
\[
(F_Y^n)^*f_*\mathcal O_X=(F_Y^n)^*S_f\oplus (F_Y^n)^*Q,
\]
hence
\[
(F_Y^n)^*(f_*\mathcal O_X)^*=(F_Y^n)^*S_f^*\oplus (F_Y^n)^*Q^*
\]
[2403.15231]. The summand \((F_Y^n)^*S_f^*\) is VGG because \(S_f\) is étale trivializable and Frobenius commutes with étale covers, while \((F_Y^n)^*Q^*\) is VGG because \(Q^*\) is ample and ample bundles are VGG after Frobenius in positive characteristic by [BP1, Theorem 2.2] [2403.15231]. This yields VGG for \((f_*\mathcal O_X)^*\), and then for \(E\) by a splitting argument [2403.15231].

A notable boundary case is also emphasized. The assumption “generically smooth” excludes purely inseparable maps such as the absolute Frobenius \(F_Y:Y\to Y\) [2403.15231]. If \(Y\) has genus at least \(2\) and \(f=F_Y\), then \(\big((f_*\mathcal O_Y)/\mathcal O_Y\big)^*\) is in fact ample, which behaves differently from the separable case [2403.15231]. This shows that the hypotheses are not merely technical. A plausible implication is that, in positive characteristic, separability assumptions are structurally intertwined with the slope computations underlying the curve-level theory.

## 5. Higher-dimensional extensions and the divergence of definitions

The paper [2509.25653] treats VGG on an irreducible normal projective variety \(X\) of dimension \(n\ge 1\) with a vector bundle \(E\), and shows that several natural higher-dimensional extensions of the curve notion diverge. Write \(P(E)\) for the Grothendieck projective bundle associated to \(E\), and \(\mathcal O_{P(E)}(1)\) for its tautological line bundle. Then
\[
H^0(P(E),\mathcal O_{P(E)}(1))=H^0(X,E)
\]
[2509.25653]. The evaluation map is
\[
ev_E:H^0(X,E)\otimes \mathcal O_X\longrightarrow E,
\]
and [2509.25653] records the equivalence that for any irreducible normal projective variety \(X\) and vector bundle \(E\), the following are equivalent: \(E\) is globally generated; \(\mathcal O_{P(E)}(1)\) is globally generated on \(P(E)\); and for every closed immersion of a smooth projective curve \(i:C\to X\), the restriction \(i^*E\) is globally generated [2509.25653].

Definition 2.1 of [2509.25653] introduces multiple notions. For an integer \(m\ge 1\), \(E\) is \(m\)-VGG if for every irreducible reduced closed subscheme \(Z\subseteq X\) with \(\dim Z=m\), there exists a finite morphism \(f:Y\to X\), where \(Y\) is an irreducible normal projective variety of dimension \(m\) and \(\operatorname{Image}(f)=Z\), such that \(f^*E\) is globally generated. It is strongly VGG if it is \(n\)-VGG; curve VGG if it is \(1\)-VGG; and VGG if there exists \(N\ge 1\) such that \(\mathcal O_{P(E)}(N)\) is globally generated on \(P(E)\) [2509.25653]. The paper also defines “swept by curves” and proves that, for vector bundles on curves, all four notions coincide [2509.25653].

The central comparison theorem is Theorem 3.1:
\[
\text{strongly VGG} \Longrightarrow \text{VGG} \Longrightarrow \text{curve VGG}
\]
in arbitrary dimension [2509.25653]. The proof of “strongly VGG \(\Rightarrow\) VGG” passes to the tautological line bundle on \(P(E)\): if \(f:Y\to X\) is finite dominant and \(f^*E\) is globally generated, then the induced finite morphism \(P_Y:=P_Y(f^*E)\to P_X:=P_X(E)\) pulls back \(\mathcal O_{P_X}(1)\) to \(\mathcal O_{P_Y}(1)\), which is globally generated [2509.25653]. The line bundle case is then invoked: a line bundle is strongly VGG if and only if some tensor power is globally generated [2509.25653].

The implication “VGG \(\Rightarrow\) curve VGG” uses symmetric powers. If \(\mathcal O_{P_X}(N)\) is globally generated and \(g:C\to X\) is a finite morphism from a smooth curve, then
\[
P_C:=P_C(g^*E)=C\times_X P_X,
\qquad
\pi_*\mathcal O_{P_C}(N)=\operatorname{Sym}^N(g^*E)
\]
and \(\mathcal O_{P_C}(N)\) is globally generated as a pullback [2509.25653]. Hence \(\operatorname{Sym}^N(g^*E)\) is globally generated, which implies
\[
\mu_{\min}(\operatorname{Sym}^N(g^*E))\ge 0,
\qquad
\mu_{\min}(g^*E)\ge 0
\]
and then the curve criterion from [BP2] yields VGG for \(g^*E\) [2509.25653]. This suggests that the higher-dimensional theory still routes essential information through restrictions to curves, but no longer in a way that produces a unique extension of the curve notion.

## 6. Examples, non-equivalences, and limitations

Several examples sharply delimit what VGG can and cannot mean. In [2403.15231], the power map
\[
f:\mathbf P^1\to \mathbf P^1,\qquad t\mapsto t^n
\]
over a field of characteristic \(0\) is finite, separable, and non-étale, hence genuinely ramified because \(\pi_1^{\mathrm{\acute et}}(\mathbf P^1)\) is trivial [2403.15231]. Consequently,
\[
E=\big((f_*\mathcal O_{\mathbf P^1})/\mathcal O_{\mathbf P^1}\big)^*
\]
is ample and virtually globally generated [2403.15231]. On \(\mathbf P^1\), every vector bundle splits as a direct sum of line bundles, so \(E\) is a direct sum of positive line bundles and hence ample [2403.15231].

The opposite behavior appears for étale covers. If \(Y\) is an elliptic curve over \(k\) of characteristic \(0\) and \(f=[m]:Y\to Y\) is multiplication by \(m\ge 2\), then \(f\) is finite étale and
\[
f_*\mathcal O_Y\cong \mathcal O_Y^{\oplus m^2},
\qquad
Q\cong \mathcal O_Y^{\oplus (m^2-1)},
\qquad
E=Q^*\cong \mathcal O_Y^{\oplus (m^2-1)}
\]
[2403.15231]. Thus \(E\) is trivial, so it is not ample; nevertheless it is virtually globally generated, indeed globally generated [2403.15231]. This directly rules out the misconception that VGG implies ampleness.

The higher-dimensional paper [2509.25653] shows that even more basic identifications fail in dimension at least \(2\). Example 4.1 constructs a rank-\(2\) holomorphic vector bundle \(E\) on a smooth complex projective curve \(M\) of genus \(g\ge 2\), associated to a dense representation \(\rho:\pi_1(M,x_0)\to SU(2)\), such that \(E\) is ample and curve VGG, but for every \(d\ge 1\),
\[
H^0(M,\operatorname{Sym}^d E)=0,
\qquad
H^0(P(E),\mathcal O_{P(E)}(d))=0
\]
[2509.25653]. Hence \(E\) is not VGG in the sense defined via \(\mathcal O_{P(E)}(N)\) [2509.25653]. Example 4.2 further shows that for line bundles one can have \(m\)-VGG without \((m+1)\)-VGG, and Example 4.3 shows that “swept by curves” does not imply curve VGG [2509.25653].

The curve-specificity of [2403.15231] is underscored by an explicit higher-dimensional counterexample. Corollary 3.5, which asserts VGG for the Tschirnhausen bundle on curves, fails in higher dimensions: if \(X\) is the blow-up of \(\mathbf{CP}^2\) at a point and \(Y=X/(\mathbf Z/2\mathbf Z)\) is the quotient by a lifted involution, then for the quotient map \(f:X\to Y\), the line bundle \(\big((f_*\mathcal O_X)/\mathcal O_Y\big)^*\) is not VGG [2403.15231]. The general higher-dimensional theory in [2509.25653] is therefore not merely an extension of the curve case; it is a reorganization of several inequivalent notions that happen to coincide only when \(\dim X=1\).

An open problem stated in [2509.25653] asks whether VGG in the sense that \(\mathcal O_{P(E)}(N)\) is globally generated for some \(N\) implies strongly VGG. The answer is known to be yes for line bundles and for direct sums of line bundles, but remains unsettled in general [2509.25653]. A plausible implication is that the tautological-line-bundle formulation captures a substantial part of virtual positivity, yet does not presently subsume the full finite-cover formulation outside special cases.

Source: https://www.emergentmind.com/topics/virtual-global-generation-vgg