Bounding the signed count of real bitangents to plane quartics. (English) Zbl 1534.14032
In this article, the authors prove a conjecture by Larson and Vogt regarding the signed count of the number of real bitangents to real smooth plane quartics.
Definition. Let \(Q\) be a quartic curve in \(\mathbb{P}^2\), \(B\) be a bitangent line to \(Q\), and \(Q\cap B = 2Z\), where \(Z = z_1+z_2\) is a degree two divisor. Suppose \(L\) is a real line such that \(L\) is disjoint from the points \(z_1\) and \(z_2\). (Such a line \(L\) is called admissible). Denote by \(\partial_L\) a derivation with respect to a linear form vanishing along \(L\). The QType of \(B\) with respect to \(L\) is defined as: \[\operatorname{QType}_L(B) := \operatorname{sign}(\partial_L f(z_1) \cdot \partial_L f(z_2)))\in \{+1,-1\}.\]
For an admissible line \(L\subset \mathbb{P}^2\), \[ s_L(Q):= \sum_{B \text{ real bitangent}} \operatorname{QType}_L(B)\]
H. Larson and I. Vogt [Res. Math. Sci. 8, No. 2, Paper No. 26, 21 p. (2021; Zbl 1471.14070)] proved that \(s_L(Q)\) is nonnegative even integers and conjectured that \(s_L(Q)\) is at most \(8\).
The authors prove that \(s_L(Q)\in\{0,2,4,6,8\}\) by gathering ideas from real algebraic geometry and plane geometry.
Definition. Let \(Q\) be a quartic curve in \(\mathbb{P}^2\), \(B\) be a bitangent line to \(Q\), and \(Q\cap B = 2Z\), where \(Z = z_1+z_2\) is a degree two divisor. Suppose \(L\) is a real line such that \(L\) is disjoint from the points \(z_1\) and \(z_2\). (Such a line \(L\) is called admissible). Denote by \(\partial_L\) a derivation with respect to a linear form vanishing along \(L\). The QType of \(B\) with respect to \(L\) is defined as: \[\operatorname{QType}_L(B) := \operatorname{sign}(\partial_L f(z_1) \cdot \partial_L f(z_2)))\in \{+1,-1\}.\]
For an admissible line \(L\subset \mathbb{P}^2\), \[ s_L(Q):= \sum_{B \text{ real bitangent}} \operatorname{QType}_L(B)\]
H. Larson and I. Vogt [Res. Math. Sci. 8, No. 2, Paper No. 26, 21 p. (2021; Zbl 1471.14070)] proved that \(s_L(Q)\) is nonnegative even integers and conjectured that \(s_L(Q)\) is at most \(8\).
The authors prove that \(s_L(Q)\in\{0,2,4,6,8\}\) by gathering ideas from real algebraic geometry and plane geometry.
Reviewer: Jaewoo Jung (Daejeon)
MSC:
| 14H50 | Plane and space curves |
| 14P99 | Real algebraic and real-analytic geometry |
| 14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |
Citations:
Zbl 1471.14070Software:
RealBitangentsReferences:
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| [2] | Kummer, Mario, Totally real theta characteristics, Ann. Mat. Pura Appl., 4, 198-6, 2141-2150 (2019) · Zbl 1441.14192 · doi:10.1007/s10231-019-00858-5 |
| [3] | Kummer, M.: A signed count of 2-torsion points on real abelian varieties. arXiv preprint arXiv:2301.10621, (2023) |
| [4] | Larson, H.; Vogt, I., An enriched count of the bitangents to a smooth plane quartic curve, Res. Math. Sci., 8, 2, 26 (2021) · Zbl 1471.14070 · doi:10.1007/s40687-021-00260-9 |
| [5] | Markwig H, Payne S, Shaw K.: Bitangents to plane quartics via tropical geometry: rationality, \(A^1\)-enumeration, and real signed count. arXiv preprint arXiv:2207.01305, (2022) |
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