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 \(\mathbb {K}[s,t;u,v]\). 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.
Dickenstein is supported by ANPCyT PICT 2016-0398, UBACYT 20020170100048BA, and CONICET PIP 11220150100473, Argentina.
Schenck is supported by NSF 1818646, Fulbright FSP 5704.
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Acknowledgements
Most of this paper was written while the third author was visiting Universidad de Buenos Aires on a Fulbright grant, and he thanks the Fulbright foundation for support and his hosts for providing a wonderful visit. All computations were done using Macaulay2 [17].
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Botbol, N., Dickenstein, A., Schenck, H. (2021). The Simplest Minimal Free Resolutions in \({\mathbb {P}^1 \times \mathbb {P}^1}\) . In: Peeva, I. (eds) Commutative Algebra. Springer, Cham. https://doi.org/10.1007/978-3-030-89694-2_3
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