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Weierstrass semigroups are well known along the literature. We present a new family of non- Weierstrass semigroups which can be written as an intersection of Weierstrass semigroups. In addition, we provide methods for computing non-Weierstrass semigroups with genus as large as desired., Funding information: Part of this paper was written during a visit of Fernando Torres to the Universidad de Cadiz (Spain) ; his visit was partially supported by Ayudas para Estancias Cortas de Investigadores (EST2018-R0, Programa de Fomento e Impulso de la Investigacion y la Transferencia en la Universidad de Cadiz) . Fernando Torres was partially supported by CNPq/Brazil (Grant 310623/2017-0) . Juan Ignacio Garcia-Garcia, Daniel Marin-Aragon, and Alberto Vigneron-Tenorio were partially supported by Junta de Andalucia research groups FQM-343 and FQM-366, and by the project MTM2017-84890-P (MINECO/FEDER, UE) .

Let $$A \subset \mathbb Z$$ A ⊂ Z be a finite subset. We denote by $${{\,\mathrm{\mathcal {B}}\,}}(A)$$ B ( A ) the set of all integers $$n \ge 2$$ n ≥ 2 such that $$|nA| > (2n-1)(|A|-1),$$ | n A | > ( 2 n - 1 ) ( | A | - 1 ) , where $$nA=A+\cdots +A$$ n A = A + ⋯ + A denotes the n-fold sumset of A. The motivation to consider $${{\,\mathrm{\mathcal {B}}\,}}(A)$$ B ( A ) stems from Buchweitz’s discovery in 1980 that if a numerical semigroup $$S \subseteq \mathbb N$$ S ⊆ N is a Weierstrass semigroup, then $${{\,\mathrm{\mathcal {B}}\,}}(\mathbb N{\setminus } S) = \emptyset .$$ B ( N \ S ) = ∅ . By constructing instances where this condition fails, Buchweitz disproved a longstanding conjecture by Hurwitz (Math Ann 41:403–442, 1893). In this paper, we prove that for any numerical semigroup $$S \subset \mathbb N$$ S ⊂ N of genus $$g \ge 2,$$ g ≥ 2 , the set $${{\,\mathrm{\mathcal {B}}\,}}(\mathbb N{\setminus } S) $$ B ( N \ S ) is finite, of unbounded cardinality as S varies.