Maximal Jumping Lines in Logarithmic Bundles

• Jumping lines of maximal jumping order are lines in the projective plane where the logarithmic bundle’s splitting deviates most from the generic type.
• The approach constructs interpolation matrices whose determinantal loci exactly capture the degeneracy and jet conditions leading to unexpected curves.
• Explicit examples and geometric interpretations show that maximal jumping order effectively quantifies extreme syzygy behavior for finite point arrangements.

Jumping lines of maximal jumping order arise in the study of logarithmic bundles (syzygy bundles) on the projective plane, and are identified as lines along which the splitting type of the bundle deviates most significantly from the generic case. Their analysis connects interpolation matrices, degeneracy loci, and phenomena of unexpected curves in algebraic geometry, providing determinantal loci that measure maximal deviation in bundle behavior associated to finite point arrangements in $\mathbb{P}^2$ (Guardo et al., 16 Jan 2026).

1. Logarithmic Bundles and Splitting Deviations

Let $Z = \{P_1,\ldots,P_n\} \subset \mathbb{P}^2$ denote a finite set of distinct points, and let $A_Z \subset (\mathbb{P}^2)^\vee$ be the dual arrangement of lines $\ell_i$ corresponding to each $P_i$. The product $f = \prod_{i=1}^n \ell_i$ yields the Jacobian ideal $$J_f = (f_x, f_y, f_z) \subset S = \mathbb{C}[x, y, z]$$. The syzygy module

$$AR(f) = \{(a,b,c) \in S^3 : a f_x + b f_y + c f_z = 0\}$$

sheafifies, after twisting, to the logarithmic bundle $E_Z = \widetilde{AR(f)}(1-d)$ where $d = n = \deg f$. Alternatively, $E_Z$ is the kernel of the map

$$\mathcal{O}_{\mathbb{P}^2}^{\oplus 3} \xrightarrow{(f_x, f_y, f_z)} \mathcal{O}_{\mathbb{P}^2}(d-1).$$

Restricting a rank-2 bundle $E$ to a line $L \simeq \mathbb{P}^1$ yields by Grothendieck's theorem

$$E|_L \simeq \mathcal{O}_L(a(L)) \oplus \mathcal{O}_L(b(L)),$$

where $a(L) \ge b(L)$ and $a(L) + b(L) = c_1(E)$. For a generic line, the splitting is $(a_0, b_0)$. A line $L$ is a jumping line for $E$ if its splitting type differs, i.e. $(a(L), b(L)) \ne (a_0, b_0)$. The jumping order is defined by $\jmath_E(L) := a(L)-a_0 \ge 0$; the lines with maximal jumping order form the set

$$\jmath_{\max}(E) := \max_{L \subset \mathbb{P}^2} \jmath_E(L).$$

2. Interpolation Matrices: Construction and Role

Fixing nonnegative integers $d \ge m \ge 1$ and $Z = \{P_1,\ldots,P_s\}$, consider $S_d$ as the space of degree-$d$ homogeneous forms with basis $w = \{w_1,\ldots,w_N\}$, $N = \binom{d+2}{2}$. Selecting a general point $B = (a_0 : a_1 : a_2) \in \mathbb{P}^2$ yields the interpolation matrix $M(d, m; Z; B)$, an $N \times N$ matrix:

• Rows $i = 1, \ldots, s$: $M_{i,j} = w_j(p_{i0}, p_{i1}, p_{i2})$ for each $P_i = (p_{i0}:p_{i1}:p_{i2})$,
• Rows $i = s+1, \ldots, N$: For each multi-index $\alpha = (\alpha_0, \alpha_1, \alpha_2)$ of $|\alpha| = m-1$, $$M_{s + \mu, j} = \frac{\partial^\alpha w_j}{\partial x^{\alpha_0} \partial y^{\alpha_1} \partial z^{\alpha_2}} \big|_{(x,y,z) = (a_0, a_1, a_2)}$$.

When $s + \binom{m+1}{2} = N$, this is termed the "square case". The determinant

$F_{d,m;Z}(B) := \det M(d,m;Z;B)$

is bihomogeneous and vanishes exactly on the locus of interest in $(\mathbb{P}^2)^\vee$.

3. Determinantal Loci and First-Kind Jumping Lines

With appropriate choices of $(d,m)$ and $Z$, Theorem 3.1 asserts that the zero set of $F_{d,m;Z}(B)$ precisely coincides with the locus $V_1(E_Z)$ of jumping lines of first kind:

$$V_1(E_Z) = \{ B \in (\mathbb{P}^2)^\vee : F_{d,m;Z}(B) = 0 \},$$

where $E_Z$ is normalized such that $c_1(E_Z) \in \{0, -1\}$. The degeneracy locus of the evaluation map

$$S_d \to \mathbb{C}^s \oplus \mathbb{C}^{\binom{m+1}{2}}$$

is thus captured scheme-theoretically by $\{F_{d,m;Z} = 0\}$.

When $Z$ is general and $|Z| = 2d+1$, $c_1(E_Z) = 0$ and

$\deg V_1(E_Z) = c_2(E_Z) = d(d-1),$

with $F_{d,d-1;Z}$ irreducible and lacking fixed components.

4. Multiplicities and Maximal Jumping Order

A fundamental fact is that the jumping order $\jmath_E(L)$ of a line $L$ coincides with the vanishing order $\operatorname{mult}_B F_{d,m;Z}$ at its dual point $B$: the multiplicity of $F_{d,m;Z}$ at $B$ equals $\jmath_E(L)$. Higher multiplicity points in the determinantal locus correspond to lines of maximal jumping order.

The maximal jumping order is thus

$$\jmath_{\max}(E_Z) = \max_{B \in (\mathbb{P}^2)^\vee} \operatorname{mult}_B F_{d,m;Z}.$$

If an irreducible curve $C \subset \mathbb{P}^2$ of degree $t$ exists such that forms in $L(d; jB + Z)$ always have $C$ as fixed component for $j \ge j_0$, then for every $B \in C$, the multiplicity satisfies

$$\mu_C := |Z \cap C| + j_0 - td + \frac{t^2 - 3t}{2},$$

providing a lower bound for jumping order along dual lines tangent or secant to $C$.

5. Explicit Examples

For $Z = \{P_1,P_2,P_3\}$, three non-collinear points, taking $(d,m) = (2,2)$ leads to a $6 \times 6$ interpolation matrix with basis elements $\{x^2, y^2, z^2, xy, xz, yz\}$ and explicit jet conditions at $B$. Its determinant becomes

$$F_{2,2;Z}(a_0:a_1:a_2) = c \prod_{1 \le i < j \le 3} \ell_{ij}(B),$$

with each jumping line (secant) appearing with multiplicity 1; thus, $\jmath_{\max} = 1$.

For $Z$ as five points in general position, $(d,m) = (2,1)$ yields again an irreducible determinant of degree 2: all jumping lines correspond to tangents to the unique conic passing doubly through a point in $Z$, and again $\jmath_{\max} = 1$.

6. Geometric Frameworks and Interpretations

The determinantal construction of $F_{d,m;Z}$ generalizes classical jumping-line loci in several frameworks:

• In the Dolgachev-Kapranov setup, the first-kind jumping curve for $T\langle A \rangle(1)$ is the branch locus of the polar map; the present approach yields an explicit determinantal locus for point-duals.
• In Barth’s stable rank-2 bundle theory with $c_1 = 0$, the jumping curve of degree $c_2$ combined with theta-characteristics suffices to determine the bundle; the interpolation determinant affords a direct construction.
• In the context of unexpected curves, $F_{d,m;Z}(B) = 0$ encodes the existence of degree-$d$ curves through $Z$ with a fat point of multiplicity $m$ at $B$, a phenomenon regarded as "unexpected" when $\binom{d + 2}{2} - |Z| - \binom{m + 1}{2} = 0$.

A plausible implication is that maximal jumping order identifies the most extreme unexpected behavior for linear systems of plane curves passing through $Z$, corresponding to high multiplicity components of the determinantal locus.

7. Significance and Applications

Jumping lines of maximal jumping order furnish a powerful tool for analyzing degeneracy phenomena in syzygy bundles associated with finite point arrangements, supplying explicit determinantal equations for loci whose geometric and combinatorial features reflect both classical bundle theory and the algebraic structure of plane curves containing imposed singularities. Their structure encodes both local splitting deviations and global dependencies on arrangements, bridging interpolation conditions, vector bundle splitting, and unexpected geometric behavior (Guardo et al., 16 Jan 2026).