### Arithmetic properties of monomial curves

Coordinator: Santiago Zarzuela Armengou (Universitat de Barcelona)

**Members**

- Isabel Bermejo Díaz (Universidad de La Laguna)
- Gemma Colomé Nin (Universitat Pompeu Fabra)
- Teresa Cortadellas Benítez (Universitat de Barcelona i Universitat Pompeu Fabra)
- Juan Elías García (Universitat de Barcelona)
- Ignacio García Marco (Université Aix-Maresille)
- Philippe Giménez (Universidad de Valladolid)
- Roser Homs Pons (Universitat de Barcelona)
- Raheleh Jafari (Mosaheb Institute of Mathematics, Kharazmi University, Tehran, Irán)
- Anna Oneto (Universidad de Génova, Italia)
- Francesc Planas Vilanova (Universitat de Catalunya)
- Francesco Strazzanti (Universidad de Pisa, Italia; Universidad de Sevilla)
- Grazia Tamone (Universidad de Génova, Italia)
- Rafael Villareal (Cinestav, Instituto Politécnico Nacional, Méjico)

**Description of the research activity**

The activities of this node turn around the study of the arithmetical properties of monomial curves. Any numerical semigroup provides the parametrization of a monomial curve whose coordinate ring is given by the corresponding numerical semigroup ring. In this way it is stablished a fruitful correspondence between the arithmetical properties of the monomial curve (those of its coordinate ring) and the properties of the numerical semigroup. On the other side, the study of monomial curves is a very rich and active research area, both from the geometric and the algebraic point of view. Their complexity but at the same realizable study make them a paradigmatic case where to look and verify more general statements, for affine curves or for higher dimensional affine varieties. The members of the node are among the most active international researchers of some of the fundamental aspects of monomial curves. Namely, they have made outstanding contributions to the behavior of the Hilbert function, to the characterization of the arithmetical properties of the tangent cone of monomial curves (Cohen-Macaulay, Gorenstein, Buchsbaum, complete intersection), to the behavior of those properties by gluing, to the problem of determining the minimal number of generators of the defining ideal of the curve, to the explicit computation of the free resolution of such defining ideal, to the behavior of this resolution by shifting, and to the computation of its Castelnuovo-Mumford regularity. The methods are based by one hand on the direct study of the numerical semigroup, specially by means of its Apéry basis, and on the other hand on a detailed study of the definition ideal of the curve as a toric ideal, and so by using the combinatorial methods usually available to study such kind of ideals.