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Título
Projective Cohen-Macaulay monomial curves and their affine charts
Año del Documento
2025
Editorial
Springer
Descripción
Producción Científica
Documento Fuente
Ricerche di Matematica, 2025.
Zusammenfassung
In this paper, we explore when the Betti numbers of the coordinate rings of a projective
monomial curve and one of its affine charts are identical. Given an infinite field k
and a sequence of relatively prime integers a0 = 0 < a1 < · · · < a n = d, we
consider the projective monomial curve C ⊂ P n
k of degree d parametrically defined
by x i = u ai vd−ai for all i ∈ {0, . . . , n} and its coordinate ring k[C]. The curve
C1 ⊂ An
k with parametric equations x i = t ai for i ∈ {1, . . . , n} is an affine chart
of C and we denote by k[C1] its coordinate ring. The main contribution of this paper
is the introduction of a novel (Gröbner-free) combinatorial criterion that provides a
sufficient condition for the equality of the Betti numbers of k[C] and k[C1]. Leveraging
this 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 different
values of j ∈ N. In this context, Vu proved that for large enough values of j, one
has an equality between the Betti numbers of the corresponding affine and projective
curves. Using our results, we improve Vu’s upper bound for the least value of j such
that this occurs.
Materias Unesco
12 Matemáticas
Palabras Clave
Projective monomial curve
Affine monomial curve
Apery set
Poset
Betti numbers
ISSN
0035-5038
Revisión por pares
SI
Patrocinador
Open access funding provided by FEDER European Funds and the Junta De Castilla y León under the Research and Innovation Strategy for Smart Specialization (RIS3) of Castilla y León 2021-2027.
This work was supported in part by Ministerio de Ciencia, Innovación y Universidades and by ERDF/EU (grant PID2022-137283NB-C22)
This work was supported in part by Ministerio de Ciencia, Innovación y Universidades and by ERDF/EU (grant PID2022-137283NB-C22)
Version del Editor
Propietario de los Derechos
© 2025 The Author(s)
Idioma
eng
Tipo de versión
info:eu-repo/semantics/publishedVersion
Derechos
openAccess
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