On explicit birational geometry for polarised varieties

2023/03/22 by Minzhe Zhu, Zhu, Minzhe · 2 citations
Mathematics · Social Sciences · #14J32 #14J40 #14J45 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Vietnamese History and Culture Studies

paper · pdf · doi:10.48550/arxiv.2303.12292

openalex publication_date 2023/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we investigate the explicit birational geometry for projective ε-lc varieties polarised by nef and big Weil divisors. We show that if X is a projective ε-lc variety, H is a nef and big Weil divisor with dimφ_H(X)≥ n-1 and L is an effective Weil divisor such that |L-K_X|≠ ∅ or L-K_X is nef, then we can find an explicit lower bound of vol(H) and prove that |L+m^′ H| is birational for m^′≥ m, where m is an explicit number which depends only on n and ε. This result can be applied to polarised Calabi-Yau varieties, Fano varieties and varieties of general type, generalizing the results in [CEW22] and [Zhu23].

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• Explicit bounds on foliated surfaces and the Poincaré problem
• Explicit Effective Birationality for Singular Surfaces

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