Virtual Global Generation (VGG)

• Virtual Global Generation (VGG) is a relaxation of classical global generation for vector bundles, using finite pullback from smooth curves to guarantee global sections.
• It distinguishes between ampleness and global generation, as seen in Tschirnhausen bundles that are ample only when the underlying morphism is genuinely ramified.
• Extensions to higher dimensions reveal multiple non-equivalent VGG notions, underscoring the unique role of curve methods in understanding vector bundle positivity.

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 (Biswas et al., 30 Sep 2025). 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 (Biswas et al., 30 Sep 2025). 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 (Biswas et al., 2024).

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$0 be a vector bundle on $E$1. Global generation means that the natural evaluation map

$E$2

is surjective (Biswas et al., 2024). Following Section 3 of (Biswas et al., 2024), $E$3 is virtually globally generated if there exists a finite surjective morphism $E$4 from an irreducible smooth projective curve $E$5 such that $E$6 is generated by its global sections. The same paper also uses the related notion “étale trivializable”: $E$7 is étale trivializable if there exists an étale cover $E$8 with $E$9 trivial (Biswas et al., 2024).

The curve definition used in (Biswas et al., 30 Sep 2025) is equivalent in spirit: if $f:Y\to X$0 is a smooth projective curve and $f:Y\to X$1 a vector bundle on $f:Y\to X$2, then $f:Y\to X$3 is VGG if there exists a nonconstant morphism $f:Y\to X$4 from a smooth irreducible projective curve $f:Y\to X$5 such that $f:Y\to X$6 is globally generated (Biswas et al., 30 Sep 2025). 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 $f:Y\to X$7, then $f:Y\to X$8 is VGG if and only if $f:Y\to X$9 is a direct sum of a finite vector bundle and an ample vector bundle; if $Y$0, then $Y$1 is VGG if and only if $Y$2 is a sum of a finite vector bundle and an ample vector bundle for some $Y$3, where $Y$4 is the absolute Frobenius (Biswas et al., 30 Sep 2025).

A standard geometric source of VGG in (Biswas et al., 2024) starts with a generically smooth morphism

$Y$5

between irreducible smooth projective curves over an algebraically closed field of arbitrary characteristic. Here “generically smooth” means that $Y$6 is smooth over a dense open subset of $Y$7; for morphisms of curves this is equivalent to $Y$8 being separable on the generic point (Biswas et al., 2024). Since $Y$9 and $f^*E$0 are projective curves and $f^*E$1 is nonconstant, $f^*E$2 is finite, and because the source and target are smooth curves, $f^*E$3 is a locally free $f^*E$4-sheaf (Biswas et al., 2024).

2. Pushforward algebras and the Tschirnhausen bundle

The inclusion $f^*E$5 induced by the unit of the $f^*E$6-algebra structure yields a short exact sequence

$f^*E$7

where

$f^*E$8

(Biswas et al., 2024). The dual bundle

$f^*E$9

is the Tschirnhausen bundle associated to $E:=\big((f_*\mathcal O_X)/\mathcal O_Y\big)^*$0 (Biswas et al., 2024).

If $E:=\big((f_*\mathcal O_X)/\mathcal O_Y\big)^*$1 is finite flat of degree $E:=\big((f_*\mathcal O_X)/\mathcal O_Y\big)^*$2, then

$E:=\big((f_*\mathcal O_X)/\mathcal O_Y\big)^*$3

(Biswas et al., 2024). The bundle $E:=\big((f_*\mathcal O_X)/\mathcal O_Y\big)^*$4 carries a natural $E:=\big((f_*\mathcal O_X)/\mathcal O_Y\big)^*$5-algebra structure, with $E:=\big((f_*\mathcal O_X)/\mathcal O_Y\big)^*$6 as the unit subalgebra (Biswas et al., 2024). 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 (Biswas et al., 2024): there is a short exact sequence

$E:=\big((f_*\mathcal O_X)/\mathcal O_Y\big)^*$7

where $E:=\big((f_*\mathcal O_X)/\mathcal O_Y\big)^*$8 is étale trivializable and $E:=\big((f_*\mathcal O_X)/\mathcal O_Y\big)^*$9 is ample. Here $f:X\to Y$0 is identified with the maximal semistable subbundle $f:X\to Y$1; it has degree $f:X\to Y$2 and arises from an étale factorization $f:X\to Y$3 and $f:X\to Y$4 such that $f:X\to Y$5 and $f:X\to Y$6 (Biswas et al., 2024). The quotient is then $f:X\to Y$7, and the dual positivity of this quotient is inherited from the genuinely ramified part of the factorization (Biswas et al., 2024).

This decomposition is the mechanism behind VGG in the curve case. In characteristic $f:X\to Y$8, the factorization $f:X\to Y$9 with $f$0 genuinely ramified gives

$f$1

where $f$2 is the “trace-zero” summand, and after applying $f$3 one gets

$f$4

(Biswas et al., 2024). Since $f$5 is étale trivializable and $f$6 is ample, both summands are VGG, hence $f$7 is VGG; because $f$8 is a direct summand, $f$9 is also VGG (Biswas et al., 2024).

3. Genuine ramification, slopes, and ampleness

The notion of genuine ramification is central in (Biswas et al., 2024). A dominant generically smooth morphism $Y$0 of smooth projective curves is genuinely ramified if $Y$1 is the unique maximal semistable subsheaf of $Y$2 (Biswas et al., 2024). The paper records equivalent conditions: the maximal semistable subsheaf of $Y$3 is $Y$4; $Y$5; the fiber product $Y$6 is connected; the induced homomorphism

$Y$7

is surjective; and $Y$8 does not factor through any nontrivial finite étale covering of $Y$9 (Biswas et al., 2024). In the terminology of [CLV], such an $k$0 is “primitive,” and the paper notes that “genuinely ramified” is equivalent to “primitive” (Biswas et al., 2024).

The principal ampleness statement is Corollary 3.2: for a generically smooth morphism $k$1 between irreducible smooth projective curves,

$k$2

(Biswas et al., 2024). The slope-theoretic input begins with the identity

$k$3

(Equation (2.2)), because $k$4 gives a semistable subbundle of slope $k$5, while general bounds imply $k$6 (Biswas et al., 2024).

If $k$7 is genuinely ramified, then the uniqueness of $k$8 as maximal semistable subbundle forces

$k$9

(Biswas et al., 2024). In characteristic $E$00, ampleness of a vector bundle $E$01 on a curve is equivalent to the property that every nonzero quotient line bundle of $E$02 has positive degree, and $E$03 implies ampleness (Biswas et al., 2024). In characteristic $E$04, 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 (Biswas et al., 2024).

Conversely, if $E$05 is not genuinely ramified, then the maximal semistable subbundle $E$06 has degree $E$07 and rank at least $E$08, and $E$09 is a quotient of $E$10 of degree $E$11 (Biswas et al., 2024). A vector bundle on a curve with a quotient of degree $E$12 cannot be ample, so $E$13 is not ample (Biswas et al., 2024). 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 (Biswas et al., 2024).

4. The VGG theorem for curves and Frobenius methods

The main VGG statement in (Biswas et al., 2024) is Theorem 3.3: if $E$14 is a generically smooth morphism between irreducible smooth projective curves, then $E$15 is virtually globally generated. Corollary 3.5 then gives the corresponding statement for the Tschirnhausen bundle: $E$16 is virtually globally generated (Biswas et al., 2024).

In characteristic $E$17, the proof combines the decomposition

$E$18

with two inputs: $E$19 is étale trivializable, hence $E$20 is VGG, and $E$21 is ample, while ample bundles on curves are VGG by BP2, Theorem 3.6. The same decomposition also shows that $E$22 is VGG, since it is a direct summand (Biswas et al., 2024).

In positive characteristic, the argument passes to Frobenius pullback. Let $E$23 be the absolute Frobenius. For $E$24,

$E$25

hence

$E$26

(Biswas et al., 2024). The summand $E$27 is VGG because $E$28 is étale trivializable and Frobenius commutes with étale covers, while $E$29 is VGG because $E$30 is ample and ample bundles are VGG after Frobenius in positive characteristic by BP1, Theorem 2.2. This yields VGG for $E$31, and then for $E$32 by a splitting argument (Biswas et al., 2024).

A notable boundary case is also emphasized. The assumption “generically smooth” excludes purely inseparable maps such as the absolute Frobenius $E$33 (Biswas et al., 2024). If $E$34 has genus at least $E$35 and $E$36, then $E$37 is in fact ample, which behaves differently from the separable case (Biswas et al., 2024). 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 (Biswas et al., 30 Sep 2025) treats VGG on an irreducible normal projective variety $E$38 of dimension $E$39 with a vector bundle $E$40, and shows that several natural higher-dimensional extensions of the curve notion diverge. Write $E$41 for the Grothendieck projective bundle associated to $E$42, and $E$43 for its tautological line bundle. Then

$E$44

(Biswas et al., 30 Sep 2025). The evaluation map is

$E$45

and (Biswas et al., 30 Sep 2025) records the equivalence that for any irreducible normal projective variety $E$46 and vector bundle $E$47, the following are equivalent: $E$48 is globally generated; $E$49 is globally generated on $E$50; and for every closed immersion of a smooth projective curve $E$51, the restriction $E$52 is globally generated (Biswas et al., 30 Sep 2025).

Definition 2.1 of (Biswas et al., 30 Sep 2025) introduces multiple notions. For an integer $E$53, $E$54 is $E$55-VGG if for every irreducible reduced closed subscheme $E$56 with $E$57, there exists a finite morphism $E$58, where $E$59 is an irreducible normal projective variety of dimension $E$60 and $E$61, such that $E$62 is globally generated. It is strongly VGG if it is $E$63-VGG; curve VGG if it is $E$64-VGG; and VGG if there exists $E$65 such that $E$66 is globally generated on $E$67 (Biswas et al., 30 Sep 2025). The paper also defines “swept by curves” and proves that, for vector bundles on curves, all four notions coincide (Biswas et al., 30 Sep 2025).

The central comparison theorem is Theorem 3.1: $E$68 in arbitrary dimension (Biswas et al., 30 Sep 2025). The proof of “strongly VGG $E$69 VGG” passes to the tautological line bundle on $E$70: if $E$71 is finite dominant and $E$72 is globally generated, then the induced finite morphism $E$73 pulls back $E$74 to $E$75, which is globally generated (Biswas et al., 30 Sep 2025). The line bundle case is then invoked: a line bundle is strongly VGG if and only if some tensor power is globally generated (Biswas et al., 30 Sep 2025).

The implication “VGG $E$76 curve VGG” uses symmetric powers. If $E$77 is globally generated and $E$78 is a finite morphism from a smooth curve, then

$E$79

and $E$80 is globally generated as a pullback (Biswas et al., 30 Sep 2025). Hence $E$81 is globally generated, which implies

$E$82

and then the curve criterion from [BP2] yields VGG for $E$83 (Biswas et al., 30 Sep 2025). 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 (Biswas et al., 2024), the power map

$E$84

over a field of characteristic $E$85 is finite, separable, and non-étale, hence genuinely ramified because $E$86 is trivial (Biswas et al., 2024). Consequently,

$E$87

is ample and virtually globally generated (Biswas et al., 2024). On $E$88, every vector bundle splits as a direct sum of line bundles, so $E$89 is a direct sum of positive line bundles and hence ample (Biswas et al., 2024).

The opposite behavior appears for étale covers. If $E$90 is an elliptic curve over $E$91 of characteristic $E$92 and $E$93 is multiplication by $E$94, then $E$95 is finite étale and

$E$96

(Biswas et al., 2024). Thus $E$97 is trivial, so it is not ample; nevertheless it is virtually globally generated, indeed globally generated (Biswas et al., 2024). This directly rules out the misconception that VGG implies ampleness.

The higher-dimensional paper (Biswas et al., 30 Sep 2025) shows that even more basic identifications fail in dimension at least $E$98. Example 4.1 constructs a rank-$E$99 holomorphic vector bundle $f:Y\to X$00 on a smooth complex projective curve $f:Y\to X$01 of genus $f:Y\to X$02, associated to a dense representation $f:Y\to X$03, such that $f:Y\to X$04 is ample and curve VGG, but for every $f:Y\to X$05,

$f:Y\to X$06

(Biswas et al., 30 Sep 2025). Hence $f:Y\to X$07 is not VGG in the sense defined via $f:Y\to X$08 (Biswas et al., 30 Sep 2025). Example 4.2 further shows that for line bundles one can have $f:Y\to X$09-VGG without $f:Y\to X$10-VGG, and Example 4.3 shows that “swept by curves” does not imply curve VGG (Biswas et al., 30 Sep 2025).

The curve-specificity of (Biswas et al., 2024) 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 $f:Y\to X$11 is the blow-up of $f:Y\to X$12 at a point and $f:Y\to X$13 is the quotient by a lifted involution, then for the quotient map $f:Y\to X$14, the line bundle $f:Y\to X$15 is not VGG (Biswas et al., 2024). The general higher-dimensional theory in (Biswas et al., 30 Sep 2025) is therefore not merely an extension of the curve case; it is a reorganization of several inequivalent notions that happen to coincide only when $f:Y\to X$16.

An open problem stated in (Biswas et al., 30 Sep 2025) asks whether VGG in the sense that $f:Y\to X$17 is globally generated for some $f:Y\to X$18 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 (Biswas et al., 30 Sep 2025). 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.