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Mathematics > Algebraic Geometry

arXiv:1805.07001 (math)
[Submitted on 18 May 2018 (v1), last revised 30 Apr 2020 (this version, v4)]

Title:Rational curves in holomorphic symplectic varieties and Gromov-Witten invariants

Authors:Georg Oberdieck, Junliang Shen, Qizheng Yin
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Abstract:We use Gromov-Witten theory to study rational curves in holomorphic symplectic varieties. We present a numerical criterion for the existence of uniruled divisors swept out by rational curves in the primitive curve class of a very general holomorphic symplectic variety of $K3^{[n]}$ type. We also classify all rational curves in the primitive curve class of the Fano variety of lines in a very general cubic $4$-fold, and prove the irreducibility of the corresponding moduli space. Our proofs rely on Gromov-Witten calculations by the first author, and in the Fano case on a geometric construction of Voisin. In the Fano case a second proof via classical geometry is sketched.
Comments: Minor changes. Replace Proposition 3.2 in v3, whose proof contains a gap, by Lemma 3.2. Main theorems 0.1 and 0.2 remain unchanged. Modifications comparing to v3 are summarized in Footnote 3 (Page 4)
Subjects: Algebraic Geometry (math.AG)
Cite as: arXiv:1805.07001 [math.AG]
  (or arXiv:1805.07001v4 [math.AG] for this version)
  https://doi.org/10.48550/arXiv.1805.07001
Journal reference: Adv. Math. 357 (2019) 106829

Submission history

From: Junliang Shen [view email]
[v1] Fri, 18 May 2018 00:38:49 UTC (24 KB)
[v2] Sat, 2 Jun 2018 07:24:15 UTC (26 KB)
[v3] Tue, 24 Sep 2019 17:04:34 UTC (27 KB)
[v4] Thu, 30 Apr 2020 13:46:31 UTC (28 KB)
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