The Simplest Minimal Free Resolutions in ℙ 1 × ℙ 1 $${\mathbb {P}^1 \times \mathbb {P}^1}$$

Nicolás Botbol (), Alicia Dickenstein () and Hal Schenck ()
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Nicolás Botbol: FCEN, Universidad de Buenos Aires, Ciudad Universitaria, Pab. I, Departamento de Matemática
Alicia Dickenstein: FCEN, Universidad de Buenos Aires, and IMAS (UBA-CONICET), Ciudad Universitaria, Pab. I, Departamento de Matemática
Hal Schenck: Auburn University, Department of Mathematics

A chapter in Commutative Algebra, 2021, pp 113-145 from Springer

Abstract: Abstract We study the minimal bigraded free resolution of an ideal with three generators of the same bidegree, contained in the bihomogeneous maximal ideal 〈s, t〉∩〈u, v〉 of the bigraded ring 𝕂 [ s , t ; u , v ] $$\mathbb {K}[s,t;u,v]$$ . Our analysis involves tools from algebraic geometry (Segre-Veronese varieties), classical commutative algebra (Buchsbaum-Eisenbud criteria for exactness, Hilbert-Burch theorem), and homological algebra (Koszul homology, spectral sequences). We treat in detail the case in which the bidegree is (1, n). We connect our work to a conjecture of Fröberg–Lundqvist on bigraded Hilbert functions, and close with a number of open problems.

Keywords: Bihomogeneous ideal; Syzygy; Free resolution; Segre map; Primary 14M25; Secondary 14F17 (search for similar items in EconPapers)
Date: 2021
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-030-89694-2_3

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DOI: 10.1007/978-3-030-89694-2_3

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