A numerical transcendental method in algebraic geometry

Pierre Lairez and Emre Sertöz

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Submission date: 28. Nov. 2018
published in: SIAM journal on applied algebra and geometry, 3 (2019) 4, p. 559-584 
DOI number (of the published article): 10.1137/18M122861X
Bibtex
MSC-Numbers: 32J25, 14Q10, 14C22, 32G20
Keywords and phrases: Transcendental methods, Hodge theory, algebraic geometry, Picard groups, Period matrices, Variation of Hodge structure
Link to arXiv: See the arXiv entry of this preprint.

Abstract:
Based on high precision computation of periods and lattice reduction techniques, we compute the Picard group of smooth surfaces. We also study the lattice reduction technique that is employed in order to quantify the possibility of numerical error in terms of an intrinsic measure of complexity of each surface. The method applies more generally to the computation of the lattice generated by Hodge cycles of middle dimension on smooth projective hypersurfaces. We demonstrate the method by a systematic study of thousands of quartic surfaces (K3s) defined by sparse polynomials. As an application, we count the number of rational curves of a given degree lying on each surface. For quartic surfaces we also compute the endomorphism ring of their transcendental lattice.