Modular Frobenius pseudo-varieties

If $$m \in {\mathbb {N}} \setminus \{0,1\}$$ m ∈ N \ { 0 , 1 } and A is a finite subset of $$\bigcup _{k \in {\mathbb {N}} \setminus \{0,1\}} \{1,\ldots ,m-1\}^k$$ ⋃ k ∈ N \ { 0 , 1 } { 1 , … , m - 1 } k , then we denote by $$\begin{aligned} {\mathscr {C}}(m,A) =&\{ S\in {\mathscr {S}}_m \mid s_1+\cdots +s_k-m \in S \text { if } (s_1,\ldots ,s_k)\in S^k \text { and } \\ {}&\qquad (s_1 \bmod m, \ldots , s_k \bmod m)\in A \}. \end{aligned}$$ C ( m , A ) = { S ∈ S m ∣ s 1 + ⋯ + s k - m ∈ S if ( s 1 , … , s k ) ∈ S k and ( s 1 mod m , … , s k mod m ) ∈ A } . In this work we prove that $${\mathscr {C}}(m,A)$$ C ( m , A ) is a Frobenius pseudo-variety. We also show algorithms that allows us to establish whether a numerical semigroup belongs to $${\mathscr {C}}(m,A)$$ C ( m , A ) and to compute all the elements of $${\mathscr {C}}(m,A)$$ C ( m , A ) with a fixed genus. Moreover, we introduce and study three families of numerical semigroups, called of second-level, thin and strong, and corresponding to $${\mathscr {C}}(m,A)$$ C ( m , A ) when $$A=\{1,\ldots ,m-1\}^3$$ A = { 1 , … , m - 1 } 3 , $$A=\{(1,1),\ldots ,(m-1,m-1)\}$$ A = { ( 1 , 1 ) , … , ( m - 1 , m - 1 ) } , and $$A=\{1,\ldots ,m-1\}^2 \setminus \{(1,1),\ldots ,(m-1,m-1)\}$$ A = { 1 , … , m - 1 } 2 \ { ( 1 , 1 ) , … , ( m - 1 , m - 1 ) } , respectively.

Quasi-Ordinarization Transform of a Numerical Semigroup

In this study, we present the notion of the quasi-ordinarization transform of a numerical semigroup. The set of all semigroups of a fixed genus can be organized in a forest whose roots are all the quasi-ordinary semigroups of the same genus. This way, we approach the conjecture on the increasingness of the cardinalities of the sets of numerical semigroups of each given genus. We analyze the number of nodes at each depth in the forest and propose new conjectures. Some properties of the quasi-ordinarization transform are presented, as well as some relations between the ordinarization and quasi-ordinarization transforms.

d-elliptic Loci in Genus 2 and 3

We consider the loci of curves of genus 2 and 3 admitting a $d$-to-1 map to a genus 1 curve. After compactifying these loci via admissible covers, we obtain formulas for their Chow classes, recovering results of Faber–Pagani and van Zelm when $d=2$. The answers exhibit quasimodularity properties similar to those in the Gromov–Witten theory of a fixed genus 1 curve; we conjecture that the quasimodularity persists in higher genus and indicate a number of possible variants.

TIGHT FIBRED KNOTS WITHOUT L-SPACE SURGERIES

We show that there exist infinitely many knots of every fixed genus $g\geq 2$ which do not admit surgery to an L-space, despite resembling algebraic knots and L-space knots in general: they are algebraically concordant to the torus knot T(2, 2g + 1) of the same genus and they are fibred and strongly quasipositive.

The Tree of Good Semigroups in $${\mathbb {N}}^2$$ and a Generalization of the Wilf Conjecture

Good subsemigroups of $${\mathbb {N}}^d$$ N d have been introduced as the most natural generalization of numerical ones. Although their definition arises by taking into account the properties of value semigroups of analytically unramified rings (for instance the local rings of an algebraic curve), not all good semigroups can be obtained as value semigroups, implying that they can be studied as pure combinatorial objects. In this work, we are going to introduce the definition of length and genus for good semigroups in $${\mathbb {N}}^d$$ N d . For $$d=2$$ d = 2 , we show how to count all the local good semigroups with a fixed genus through the introduction of the tree of local good subsemigroups of $${\mathbb {N}}^2$$ N 2 , generalizing the analogous concept introduced in the numerical case. Furthermore, we study the relationships between these elements and others previously defined in the case of good semigroups with two branches, as the type and the embedding dimension. Finally, we show that an analogue of Wilf’s conjecture fails for good semigroups in $${\mathbb {N}}^2$$ N 2 .

Structure theorems for singular minimal laminations

We apply the local removable singularity theorem for minimal laminations [W. H. Meeks III, J. Pérez and A. Ros, Local removable singularity theorems for minimal laminations, J. Differential Geom. 103 (2016), no. 2, 319–362] and the local picture theorem on the scale of topology [W. H. Meeks III, J. Pérez and A. Ros, The local picture theorem on the scale of topology, J. Differential Geom. 109 (2018), no. 3, 509–565] to obtain two descriptive results for certain possibly singular minimal laminations of {\mathbb{R}^{3}}. These two global structure theorems will be applied in [W. H. Meeks III, J. Pérez and A. Ros, Bounds on the topology and index of classical minimal surfaces, preprint 2016] to obtain bounds on the index and the number of ends of complete, embedded minimal surfaces of fixed genus and finite topology in {\mathbb{R}^{3}}, and in [W. H. Meeks III, J. Pérez and A. Ros, The embedded Calabi–Yau conjectures for finite genus, preprint 2018] to prove that a complete, embedded minimal surface in {\mathbb{R}^{3}} with finite genus and a countable number of ends is proper.