Decomposable Threefold Scrolls

• Decomposable threefold scrolls are three-dimensional projective varieties constructed as the projectivization of vector bundles that split into direct sums of line bundles over P^1 and Hirzebruch surfaces.
• Their explicit splitting criteria enable clear constructions, deformation analysis, and classification of Ulrich bundles in both rational-normal and surface-base settings.
• These scrolls function as boundary cases within Hilbert schemes and in the embedding of canonical curves, bridging the gap between decomposable and indecomposable geometric models.

Searching arXiv for recent and foundational papers on decomposable threefold scrolls and related scroll geometry. Decomposable threefold scrolls are three-dimensional scrolls whose defining vector bundle splits as a direct sum of line bundles. Two settings dominate the recent literature. In the classical rational-normal-scroll setting, a smooth threefold scroll is $S_{m,n,k}\subset \mathbb P^{g-1}$ with $m,n,k\ge 1$, naturally isomorphic to $$\mathbb P\!\big(\mathcal O_{\mathbb P^1}(m)\oplus \mathcal O_{\mathbb P^1}(n)\oplus \mathcal O_{\mathbb P^1}(k)\big)$$. In the surface-base setting, the central models are threefolds of the form $$X=\mathbb P_{\mathbb F_a}(\mathcal O_{\mathbb F_a}\oplus \mathcal O_{\mathbb F_a}(0,-b))$$, polarized by a tautological-type divisor $h=\xi+C_0+cF$. Across these settings, decomposability governs projective embeddings, Hilbert-scheme behavior, the geometry of canonical curves lying on scrolls, and the structure of Ulrich bundles and their moduli (Lara et al., 2015, Fania et al., 14 Jul 2025).

1. Geometric models and basic invariants

A rational normal scroll $S=S_{m_1,\dots,m_d}\subset \mathbb P^N$ has dimension $d$, satisfies $N=e+d-1$ with $e=m_1+\cdots+m_d$, and is swept out by a family of $(d-1)$-planes called fibers. It is smooth exactly when all $m,n,k\ge 1$0. In the smooth case one has

$m,n,k\ge 1$1

so smooth rational normal scrolls are decomposable in the standard sense. For $m,n,k\ge 1$2, this specializes to

$m,n,k\ge 1$3

with $m,n,k\ge 1$4 and canonical class

$m,n,k\ge 1$5

This is the ambient geometry used in the study of canonical models of curves on threefold scrolls (Lara et al., 2015).

For decomposable threefold scrolls over a Hirzebruch surface, the base is

$m,n,k\ge 1$6

with numerical generators $m,n,k\ge 1$7 and $m,n,k\ge 1$8 satisfying

$m,n,k\ge 1$9

The scroll is

$$\mathbb P\!\big(\mathcal O_{\mathbb P^1}(m)\oplus \mathcal O_{\mathbb P^1}(n)\oplus \mathcal O_{\mathbb P^1}(k)\big)$$0

with projection $$\mathbb P\!\big(\mathcal O_{\mathbb P^1}(m)\oplus \mathcal O_{\mathbb P^1}(n)\oplus \mathcal O_{\mathbb P^1}(k)\big)$$1, tautological class $$\mathbb P\!\big(\mathcal O_{\mathbb P^1}(m)\oplus \mathcal O_{\mathbb P^1}(n)\oplus \mathcal O_{\mathbb P^1}(k)\big)$$2, and pulled-back classes

$$\mathbb P\!\big(\mathcal O_{\mathbb P^1}(m)\oplus \mathcal O_{\mathbb P^1}(n)\oplus \mathcal O_{\mathbb P^1}(k)\big)$$3

The polarization used throughout the Ulrich-bundle theory is

$$\mathbb P\!\big(\mathcal O_{\mathbb P^1}(m)\oplus \mathcal O_{\mathbb P^1}(n)\oplus \mathcal O_{\mathbb P^1}(k)\big)$$4

and it is very ample exactly when

$$\mathbb P\!\big(\mathcal O_{\mathbb P^1}(m)\oplus \mathcal O_{\mathbb P^1}(n)\oplus \mathcal O_{\mathbb P^1}(k)\big)$$5

Under this condition, $$\mathbb P\!\big(\mathcal O_{\mathbb P^1}(m)\oplus \mathcal O_{\mathbb P^1}(n)\oplus \mathcal O_{\mathbb P^1}(k)\big)$$6 embeds as a smooth nondegenerate threefold scroll with

$$\mathbb P\!\big(\mathcal O_{\mathbb P^1}(m)\oplus \mathcal O_{\mathbb P^1}(n)\oplus \mathcal O_{\mathbb P^1}(k)\big)$$7

and

$$\mathbb P\!\big(\mathcal O_{\mathbb P^1}(m)\oplus \mathcal O_{\mathbb P^1}(n)\oplus \mathcal O_{\mathbb P^1}(k)\big)$$8

Its intersection theory is controlled by

$$\mathbb P\!\big(\mathcal O_{\mathbb P^1}(m)\oplus \mathcal O_{\mathbb P^1}(n)\oplus \mathcal O_{\mathbb P^1}(k)\big)$$9

A further structural feature is the second scroll presentation

$$X=\mathbb P_{\mathbb F_a}(\mathcal O_{\mathbb F_a}\oplus \mathcal O_{\mathbb F_a}(0,-b))$$0

which exchanges the two surface parameters and plays a direct role in the Ulrich classification (Fania et al., 14 Jul 2025).

2. Decomposability through bundle splitting on Hirzebruch surfaces

In the Hilbert-scheme literature, decomposability is expressed in terms of splitting of the rank-two bundle on the base. Over $$X=\mathbb P_{\mathbb F_a}(\mathcal O_{\mathbb F_a}\oplus \mathcal O_{\mathbb F_a}(0,-b))$$1, the relevant scrolls are

$$X=\mathbb P_{\mathbb F_a}(\mathcal O_{\mathbb F_a}\oplus \mathcal O_{\mathbb F_a}(0,-b))$$2

where

$$X=\mathbb P_{\mathbb F_a}(\mathcal O_{\mathbb F_a}\oplus \mathcal O_{\mathbb F_a}(0,-b))$$3

and $$X=\mathbb P_{\mathbb F_a}(\mathcal O_{\mathbb F_a}\oplus \mathcal O_{\mathbb F_a}(0,-b))$$4 fits into

$$X=\mathbb P_{\mathbb F_a}(\mathcal O_{\mathbb F_a}\oplus \mathcal O_{\mathbb F_a}(0,-b))$$5

with

$$X=\mathbb P_{\mathbb F_a}(\mathcal O_{\mathbb F_a}\oplus \mathcal O_{\mathbb F_a}(0,-b))$$6

The splitting threshold is explicit: if

$$X=\mathbb P_{\mathbb F_a}(\mathcal O_{\mathbb F_a}\oplus \mathcal O_{\mathbb F_a}(0,-b))$$7

then every such extension splits, so

$$X=\mathbb P_{\mathbb F_a}(\mathcal O_{\mathbb F_a}\oplus \mathcal O_{\mathbb F_a}(0,-b))$$8

If

$$X=\mathbb P_{\mathbb F_a}(\mathcal O_{\mathbb F_a}\oplus \mathcal O_{\mathbb F_a}(0,-b))$$9

then the general extension is indecomposable. In that setting, “decomposable threefold scrolls” means precisely the split case $h=\xi+C_0+cF$0 (Besana et al., 2011).

For $h=\xi+C_0+cF$1 with $h=\xi+C_0+cF$2, the analogous construction uses

$h=\xi+C_0+cF$3

where

$h=\xi+C_0+cF$4

and

$h=\xi+C_0+cF$5

with

$h=\xi+C_0+cF$6

The extension space is

$h=\xi+C_0+cF$7

and the decomposition range is again numerical: $h=\xi+C_0+cF$8 Outside that range, the split bundle persists as a limit, but the general member may be indecomposable or simple depending on $h=\xi+C_0+cF$9 (Fania et al., 2014).

These splitting criteria show that decomposable threefold scrolls sit at a sharp boundary between explicit projective-bundle models and extension-theoretic families. In the $S=S_{m_1,\dots,m_d}\subset \mathbb P^N$0 and $S=S_{m_1,\dots,m_d}\subset \mathbb P^N$1 theories, decomposable objects are not merely special examples: they are the closed, split loci around which the surrounding indecomposable families are organized.

3. Hilbert-scheme behavior and flat degenerations

For scrolls over $S=S_{m_1,\dots,m_d}\subset \mathbb P^N$2, once $S=S_{m_1,\dots,m_d}\subset \mathbb P^N$3 is very ample and

$S=S_{m_1,\dots,m_d}\subset \mathbb P^N$4

the embedding

$S=S_{m_1,\dots,m_d}\subset \mathbb P^N$5

has

$S=S_{m_1,\dots,m_d}\subset \mathbb P^N$6

with

$S=S_{m_1,\dots,m_d}\subset \mathbb P^N$7

The normal bundle is unobstructed: $S=S_{m_1,\dots,m_d}\subset \mathbb P^N$8 and there exists an irreducible component of $S=S_{m_1,\dots,m_d}\subset \mathbb P^N$9 containing $d$0 that is generically smooth of expected dimension

$d$1

Moreover, the general point of that component still parametrizes a scroll $d$2 over $d$3 with the same numerical data and the same extension format $d$4 (Besana et al., 2011).

For $d$5, the Hilbert picture becomes subtler. Under

$d$6

the tautological embedding

$d$7

produces a smooth nondegenerate threefold $d$8 with

$d$9

and again

$N=e+d-1$0

The corresponding Hilbert component is irreducible, generically smooth, and of expected dimension

$N=e+d-1$1

However, unlike the cases $N=e+d-1$2, the family of the original scrolls $N=e+d-1$3 does not fill the whole component. Its general point is a parity-adjusted candidate scroll $N=e+d-1$4, with $N=e+d-1$5 and $N=e+d-1$6, obtained from modified numerical data

$N=e+d-1$7

The identification of the component proceeds through a flat embedded degeneration

$N=e+d-1$8

passing through decomposable scrolls

$N=e+d-1$9

In this way, decomposable scrolls function as the bridge between distinct strata of the same Hilbert component (Fania et al., 2014).

A central consequence is that decomposable threefold scrolls are deformation-theoretically significant even when the general point of the component is not decomposable. They encode the accessible boundary geometry of the component and make possible explicit degenerations between different projective-bundle models.

4. Ulrich bundles, minimal complexity, and wildness

For the decomposable family

$e=m_1+\cdots+m_d$0

with very ample

$e=m_1+\cdots+m_d$1

the Ulrich theory is unusually explicit. The complete classification of $e=m_1+\cdots+m_d$2-Ulrich line bundles yields

$e=m_1+\cdots+m_d$3

If $e=m_1+\cdots+m_d$4, there are exactly two Ulrich line bundles,

$e=m_1+\cdots+m_d$5

If $e=m_1+\cdots+m_d$6, $e=m_1+\cdots+m_d$7, there are exactly four,

$e=m_1+\cdots+m_d$8

If $e=m_1+\cdots+m_d$9, there are exactly six, adding

$(d-1)$0

The same theory identifies two involutions on the Ulrich set: the usual Ulrich duality $(d-1)$1, and a second involution induced by the double scroll structure $(d-1)$2 (Fania et al., 14 Jul 2025).

Rank-two Ulrich bundles are constructed as extensions of these line bundles. Typical sequences are

$(d-1)$3

with explicitly computed extension dimensions, Chern classes, and deformation spaces. In the $(d-1)$4 regime, the component built from $(d-1)$5 and $(d-1)$6 is generically smooth, rational, and has dimension

$(d-1)$7

its general point is a slope-stable, special rank-two $(d-1)$8-Ulrich bundle with

$(d-1)$9

For the component built from $m,n,k\ge 1$00 and $m,n,k\ge 1$01, the behavior depends on $m,n,k\ge 1$02 and $m,n,k\ge 1$03: in low-parameter cases it is generically smooth of dimension $m,n,k\ge 1$04, while in other cases it may degenerate to a single polystable $m,n,k\ge 1$05-equivalence class (Fania et al., 14 Jul 2025).

Higher-rank wildness is established in two stages. One result shows that if

$m,n,k\ge 1$06

then for every $m,n,k\ge 1$07 there is a generically smooth moduli component $m,n,k\ge 1$08 of rank-$m,n,k\ge 1$09 $m,n,k\ge 1$10-Ulrich bundles with

$m,n,k\ge 1$11

whose general point is slope-stable of slope

$m,n,k\ge 1$12

Consequently, $m,n,k\ge 1$13 is geometrically $m,n,k\ge 1$14-Ulrich wild in that low-$m,n,k\ge 1$15 range (Fania et al., 14 Jul 2025). A later result sharpens the moduli picture for the same decomposable family. When $m,n,k\ge 1$16, $m,n,k\ge 1$17, $m,n,k\ge 1$18, and $m,n,k\ge 1$19, the moduli space of rank-$m,n,k\ge 1$20 $m,n,k\ge 1$21-Ulrich bundles with specified Chern classes contains a component $m,n,k\ge 1$22 that is nonempty, generically smooth, unirational for $m,n,k\ge 1$23, rational for $m,n,k\ge 1$24, and whose general point is slope-stable. Its dimension is

$m,n,k\ge 1$25

and one has

$m,n,k\ge 1$26

When $m,n,k\ge 1$27 and the stated numerical hypotheses hold, the same family remains geometrically $m,n,k\ge 1$28-Ulrich wild and still satisfies $m,n,k\ge 1$29, although the modular behavior is obstructed and the emphasis shifts from smooth components to the existence of indecomposable Ulrich bundles in all ranks (Fania et al., 4 Jun 2026).

A useful contrast comes from the broader family of threefold scrolls $m,n,k\ge 1$30 over $m,n,k\ge 1$31 defined by uniform very ample rank-two bundles $m,n,k\ge 1$32. There, for $m,n,k\ge 1$33, no Ulrich line bundles exist with respect to the tautological polarization $m,n,k\ge 1$34, and

$m,n,k\ge 1$35

whereas

$m,n,k\ge 1$36

This contrast isolates a decomposability threshold in Ulrich theory: in the decomposable surface-base family, line bundles already realize minimal complexity, while in the broader unbalanced uniform family the first genuinely Ulrich objects are rank-two bundles (Fania et al., 2023).

5. Canonical curves on smooth threefold scrolls

The geometry of decomposable threefold scrolls over $m,n,k\ge 1$37 appears naturally in the study of canonical models of singular curves. Let $m,n,k\ge 1$38 be an integral projective curve with dualizing sheaf $m,n,k\ge 1$39, and let $m,n,k\ge 1$40 be the blowup along $m,n,k\ge 1$41. The canonical model $m,n,k\ge 1$42 is defined via the complete linear system induced by $m,n,k\ge 1$43. In the threefold-scroll setting, the relevant assumption is that $m,n,k\ge 1$44 lies on a smooth rational normal threefold scroll

$m,n,k\ge 1$45

as a local complete intersection (Lara et al., 2015).

If $m,n,k\ge 1$46 is an l.c.i. curve of $m,n,k\ge 1$47-type, with

$m,n,k\ge 1$48

and if $m,n,k\ge 1$49 denotes the generic number of points cut by a fiber, then

$m,n,k\ge 1$50

For a canonical model $m,n,k\ge 1$51, substituting

$m,n,k\ge 1$52

yields numerical relations between the scroll data $m,n,k\ge 1$53 and the singularity invariants $m,n,k\ge 1$54. The resulting classification is organized by $m,n,k\ge 1$55. Among its consequences: $m,n,k\ge 1$56 forces $m,n,k\ge 1$57; $m,n,k\ge 1$58 is the borderline case where nearly Gorenstein and Kunz conditions are read off from $m,n,k\ge 1$59; $m,n,k\ge 1$60 gives

$m,n,k\ge 1$61

and $m,n,k\ge 1$62 is the tetragonal range, with two possible formulas for $m,n,k\ge 1$63 and further restrictions.

For rational monomial curves with one singular point, the semigroup-theoretic description becomes exact. If

$m,n,k\ge 1$64

is such a curve, then its canonical model lies on a threefold scroll but not on a surface scroll if and only if

$m,n,k\ge 1$65

The proof uses the semigroup of values of the singular point and the criterion that a rational monomial curve lies on a $m,n,k\ge 1$66-fold scroll exactly when its exponent set can be partitioned into $m,n,k\ge 1$67 subsets, each reorderable into an arithmetic progression with the same common difference. This result is the threefold analogue of the surface-scroll characterization of trigonal curves and shows that decomposable threefold scrolls detect tetragonality even for non-Gorenstein rational monomial curves (Lara et al., 2015).

6. Rigidity under triple-solid structures

A different perspective comes from smooth threefolds $m,n,k\ge 1$68 that are simultaneously scrolls and finite triple covers of $m,n,k\ge 1$69. If

$m,n,k\ge 1$70

has degree $m,n,k\ge 1$71 and

$m,n,k\ge 1$72

is a scroll over a smooth surface with

$m,n,k\ge 1$73

then the basic numerical identity is

$m,n,k\ge 1$74

The “obvious case” is

$m,n,k\ge 1$75

viewed as a scroll over $m,n,k\ge 1$76, for which

$m,n,k\ge 1$77

Outside this case, the geometry becomes highly restrictive: $m,n,k\ge 1$78 and the base surface $m,n,k\ge 1$79 are regular, the ramification divisor is very ample, and $m,n,k\ge 1$80 is ample and spanned (Lanteri et al., 2023).

The decisive decomposability statement is that if

$m,n,k\ge 1$81

with $m,n,k\ge 1$82 and $m,n,k\ge 1$83 ample line bundles, then one is forced back to the obvious case. Thus, in every non-obvious triple-solid scroll, the bundle $m,n,k\ge 1$84 must be indecomposable. Additional rigidity results sharpen this conclusion: if $m,n,k\ge 1$85 is a Fano threefold, then necessarily

$m,n,k\ge 1$86

and if the base surface is $m,n,k\ge 1$87, then, under the paper’s generality assumptions on the induced triple plane, the same Segre product is again the only possibility. In this context, decomposable threefold scrolls do not form a broad class; they survive only as the exceptional Segre model (Lanteri et al., 2023).

Taken together, these results show that decomposable threefold scrolls occupy two markedly different positions in current research. In scroll geometry over $m,n,k\ge 1$88 and over Hirzebruch surfaces they provide explicit, calculable ambient spaces, often with rich Ulrich-bundle theory and transparent deformation behavior. Under stronger auxiliary structures, such as triple-solid morphisms, decomposability becomes severely restricted and can collapse to a unique rigid example.