{
  "abstract": "Let $D=C_1+C_2\\subset\\mathbb{P}^2$ be a simple-normal-crossing union of smooth plane curves of degrees $1\\leqslant d_1\\leqslant d_2$. We prove an effective Second Main Theorem on a nonempty Zariski-open set of ordered pairs whenever \\[ d_1,d_2\\geqslant3, \\qquad\\text{or}\\qquad d_1=2,\\ d_2\\geqslant5, \\qquad\\text{or}\\qquad d_1=1,\\ d_2\\geqslant8. \\] 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 $\\mathcal{O}_{\\mathbb{P}^2}(3)$. 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\\bigl(N_f^{[1]}(r,C_1)+N_f^{[1]}(r,C_2)\\bigr) +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 $69$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.22714",
  "author_total": 3,
  "authors": [
    {
      "name": "Lei Hou"
    },
    {
      "name": "Pengchao Wang"
    },
    {
      "name": "Song-Yan Xie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bje6mpxx7j0cv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.22714",
  "fulltext_url": null,
  "id": "bje6mpxx7j0cv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.22714",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.CV"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7204200517",
  "pdf_url": "https://arxiv.org/pdf/2608.22714",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bje6mpxx7j0cv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2608.22714",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.22714"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Invariant two-jets and effective hyperbolicity for complements of two plane curves",
  "title_text": "Invariant two-jets and effective hyperbolicity for complements of two plane curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/bje6mpxx7j0cv",
  "venue": "arXiv",
  "versions": []
}