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

In: Commutative Algebra

Author

Listed:
  • 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)

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.

Suggested Citation

  • Nicolás Botbol & Alicia Dickenstein & Hal Schenck, 2021. "The Simplest Minimal Free Resolutions in ℙ 1 × ℙ 1 $${\mathbb {P}^1 \times \mathbb {P}^1}$$," Springer Books, in: Irena Peeva (ed.), Commutative Algebra, pages 113-145, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-89694-2_3
    DOI: 10.1007/978-3-030-89694-2_3
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