Invariant two-jets and effective hyperbolicity for complements of two plane curves
Abstract
Let D=C1+C2⊂ℙ2 be a simple-normal-crossing union of smooth plane curves of degrees 1leqslant d1leqslant d2. We prove an effective Second Main Theorem on a nonempty Zariski-open set of ordered pairs whenever d1,d2geqslant3, or d1=2, d2geqslant5, or d1=1, d2geqslant8. After intersecting with the standard very-general algebraic-hyperbolicity locus, we obtain Kobayashi hyperbolicity and hyperbolic embedding of the complement. The proof uses logarithmic invariant two-jet differentials, the corrected Demailly--El Goul zero-locus argument, and a two-component adaptation of slanted vector fields with base twist 𝒪ℙ2(3). The adapted twist three is uniform in (d1,d2): simultaneous cancellation of the two root equations requires third Taylor differences, but no seventh-order polarization. For a linear component the third-remainder space vanishes, and ten compatible mixed generators replace the generic Taylor package. A numerical phase diagram shows that exact finite computation is needed only for the six boundary pairs $(3,3),(3,4),(2,6),(2,5),(1,9),(1,8)$. Their key vanishings are certified by full-column-rank matrices over finite fields. For two cubics, a strengthened range reduces to twelve theorem-relevant degree--twist cases; two low-weight self-tests bring the computational total to $168$ character--eigenvalue blocks. The exact source, certificates, and reproduction instructions form an accompanying computational archive. In the two-cubic case the resulting explicit estimate is Tf(r)leqslant57(Nf[1](r,C1)+Nf[1](r,C2)) +o(Tf(r)) |. For a conic and a quintic one may take the explicit coefficient $45$, and for a line and an octic one may take $69$.
Related
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BibTeX
@misc{hou2026invariant,
title = {{Invariant two-jets and effective hyperbolicity for complements of two plane curves}},
author = {Lei Hou and Pengchao Wang and Song-Yan Xie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.22714},
eprint = {2608.22714},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.22714},
}
RIS
TY - UNPB
TI - Invariant two-jets and effective hyperbolicity for complements of two plane curves
AU - Lei Hou
AU - Pengchao Wang
AU - Song-Yan Xie
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2608.22714
UR - https://arxiv.org/abs/2608.22714
AB - Let D=C_1+C_2⊂ℙ^2 be a simple-normal-crossing union of smooth plane curves of degrees 1leqslant d_1leqslant d_2. We prove an effective Second Main Theorem on a nonempty Zariski-open set of ordered pairs whenever d_1,d_2geqslant3, or d_1=2, d_2geqslant5, or d_1=1, d_2geqslant8. After intersecting with the standard very-general algebraic-hyperbolicity locus, we obtain Kobayashi hyperbolicity and hyperbolic embedding of the complement. The proof uses logarithmic invariant two-jet differentials, the corrected Demailly--El Goul zero-locus argument, and a two-component adaptation of slanted vector fields with base twist 𝒪_ℙ^2(3). The adapted twist three is uniform in (d_1,d_2): simultaneous cancellation of the two root equations requires third Taylor differences, but no seventh-order polarization. For a linear component the third-remainder space vanishes, and ten compatible mixed generators replace the generic Taylor package. A numerical phase diagram shows that exact finite computation is needed only for the six boundary pairs $(3,3),(3,4),(2,6),(2,5),(1,9),(1,8)$. Their key vanishings are certified by full-column-rank matrices over finite fields. For two cubics, a strengthened range reduces to twelve theorem-relevant degree--twist cases; two low-weight self-tests bring the computational total to $168$ character--eigenvalue blocks. The exact source, certificates, and reproduction instructions form an accompanying computational archive. In the two-cubic case the resulting explicit estimate is T_f(r)leqslant57(N_f^[1](r,C_1)+N_f^[1](r,C_2)) +o(T_f(r)) |. For a conic and a quintic one may take the explicit coefficient $45$, and for a line and an octic one may take $69$.
ER -