RT info:eu-repo/semantics/article
T1 Projective Cohen-Macaulay monomial curves and their affine charts
A1 García Marco, Ignacio
A1 Giménez, Philippe Thierry
A1 González Sánchez, Mario
K1 Projective monomial curve
K1 Affine monomial curve
K1 Apery set
K1 Poset
K1 Betti numbers
K1 12 Matemáticas
AB In this paper, we explore when the Betti numbers of the coordinate rings of a projectivemonomial curve and one of its affine charts are identical. Given an infinite field kand a sequence of relatively prime integers a0 = 0 < a1 < · · · < a n = d, weconsider the projective monomial curve C ⊂ P nk of degree d parametrically definedby x i = u ai vd−ai for all i ∈ {0, . . . , n} and its coordinate ring k[C]. The curveC1 ⊂ Ank with parametric equations x i = t ai for i ∈ {1, . . . , n} is an affine chartof C and we denote by k[C1] its coordinate ring. The main contribution of this paperis the introduction of a novel (Gröbner-free) combinatorial criterion that provides asufficient condition for the equality of the Betti numbers of k[C] and k[C1]. Leveragingthis criterion, we identify infinite families of projective curves satisfying this property.Also, we use our results to study the so-called shifted family of monomial curves, i.e.,the family of curves associated to the sequences j + a1 < · · · < j + a n for differentvalues of j ∈ N. In this context, Vu proved that for large enough values of j, onehas an equality between the Betti numbers of the corresponding affine and projectivecurves. Using our results, we improve Vu’s upper bound for the least value of j suchthat this occurs.
PB Springer
SN 0035-5038
YR 2025
FD 2025
LK https://uvadoc.uva.es/handle/10324/75890
UL https://uvadoc.uva.es/handle/10324/75890
LA eng
NO Ricerche di Matematica, 2025.
NO Producción Científica
DS UVaDOC
RD 25-jul-2025