v-Numbers of symbolic power filtrations

• A, Vanmathi ^[1] ; Sarkar , Parangama ^[1]
  1. [1] Indian Institute of Technology Palakkad

     Indian Institute of Technology Palakkad

     India

     
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 77, Fasc. 2, 2026, págs. 461-481
• Idioma: inglés
• DOI: 10.1007/s13348-025-00476-w
• Texto completo no disponible (Saber más ...)
• Resumen

  • We study the asymptotic behaviour of v-number and local v-numbers of Noetherian generalized symbolic power filtrations {\mathcal {I}}=\{I_n\} in a Noetherian {\mathbb N}-graded domain and show that they are quasi-linear type. We provide sufficient conditions for the existence of the limits \lim \limits _{n\rightarrow \infty }\frac{v(I_n)}{n} and \lim \limits _{n\rightarrow \infty }\frac{v_{\mathfrak {p}}(I_n)}{n} for all {\mathfrak {p}}\in {\overline{A}}({\mathcal {I}}). We explicitly compute local v-numbers and v-numbers of symbolic powers of cover ideals of complete bipartite graphs, complete graphs, cycles, K_m^s and compare them with their Castelnuovo–Mumford regularity. For every positive integer p\ge 2, we provide an example of an unmixed bipartite graph {\mathcal {H}}_p that is not a complete multipartite graph and v(J({\mathcal {H}}_p))\ge \operatorname {bight}(I({\mathcal {H}}_p)). This answers a question of Saha in (Int Math Res Not IMRN 11:9010–9019, 2024, Question 3.12). We show that for both connected bipartite graphs and connected non-bipartite graphs, the difference between the regularity and the v-number of the cover ideals can be arbitrarily large. This strengthens and gives an alternative proof of Saha in (Int Math Res Not IMRN 11:9010-9019, 2024, Theorem 3.10).

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