Sebastián Reyes Carocca | Universidad de Chile (original) (raw)

Uploads

Papers by Sebastián Reyes Carocca

Research paper thumbnail of Weil's Galois Descent Theorem from a computational point of view

Let L/ K be a finite Galois extension and let X be an affine algebraic variety defined over L. We... more Let L/ K be a finite Galois extension and let X be an affine algebraic variety defined over L. Weil's Galois descent theorem provides necessary and sufficient conditions for X to be definable over K, that is, for the existence of an algebraic variety Y defined over K together with a birational isomorphism R:X → Y defined over L. Weil's proof does not provide a method to construct the birational isomorphism R. The aim of this paper is to give an explicit construction of R.

Research paper thumbnail of Nilpotent groups of automorphisms of families of Riemann surfaces

In this article we extend results of Zomorrodian to determine upper bounds for the order of a nil... more In this article we extend results of Zomorrodian to determine upper bounds for the order of a nilpotent group of automorphisms of a complex d-dimensional family of compact Riemann surfaces, where d ≥ 1. We provide conditions under which these bounds are sharp and, in addition, for the one-dimensional case we construct and describe an explicit family attaining the bound for infinitely many genera. We obtain similar results for the case of p-groups of automorphisms.

Research paper thumbnail of Weil’s Galois descent theorem from a computational point of view

arXiv: Algebraic Geometry, 2012

Let mathcalL/mathcalK{\mathcal L}/{\mathcal K}mathcalL/mathcalK be a finite Galois extension and let XXX be an affine algebraic v... more Let mathcalL/mathcalK{\mathcal L}/{\mathcal K}mathcalL/mathcalK be a finite Galois extension and let XXX be an affine algebraic variety defined over mathcalL{\mathcal L}mathcalL. Weil's Galois descent theorem provides necessary and sufficient conditions for XXX to be definable over mathcalK{\mathcal K}mathcalK, that is, for the existence of an algebraic variety YYY defined over mathcalK{\mathcal K}mathcalK together with a birational isomorphism R:XtoYR:X \to YR:XtoY defined over mathcalL{\mathcal L}mathcalL. Weil's proof does not provide a method to construct the birational isomorphism R.R.R. The aim of this paper is to give an explicit construction of RRR.

Research paper thumbnail of On the fiber product of Riemann surfaces II

Let S0,S1S_{0}, S_{1}S0,S1 and S2S_{2}S2 be connected Riemann surfaces and let beta1:S1toS0\beta_{1}:S_{1} \to S_{0}beta1:S1toS0 ... more Let S0,S1S_{0}, S_{1}S0,S1 and S2S_{2}S2 be connected Riemann surfaces and let beta1:S1toS0\beta_{1}:S_{1} \to S_{0}beta1:S1toS0 and beta2:S2toS0\beta_{2}:S_{2} \to S_{0}beta2:S2toS0 be surjective holomorphic maps. The associated fiber product S1times(beta1,beta2)S2S_{1} \times_{(\beta_{1},\beta_{2})} S_{2}S1times(beta1,beta2)S2 has the structure of a singular Riemann surface, endowed with a canonical map beta\betabeta to S0S_{0}S0 satisfying that betajcircpij=beta\beta_{j} \circ \pi_{j}=\betabetajcircpij=beta, where pij\pi_{j}pij is coordinate projection onto SjS_{j}Sj. In this paper we provide a Fuchsian description of the fiber product and obtain that if one the maps betaj\beta_{j}betaj is a regular branched cover, then all its irreducible components are isomorphic. In the case that both betaj\beta_{j}betaj are of finite degree, we observe that the number of irreducible components is bounded above by the greatest common divisor of the two degrees; we study the irreducibility of the fiber product. In the case that S0=widehatmathbbCS_{0}=\widehat{\mathbb C}S0=widehatmathbbC, and S1S_{1}S1 and S2S_{2}S2 are compact, we define the strong field of moduli of the pair $(S_{1} ...

Research paper thumbnail of A note on Jacobians of quasiplatonic Riemann surfaces with complex multiplication

Geometriae Dedicata

Let m 6 be an even integer. In this short note we prove that the Jacobian variety of a quasiplato... more Let m 6 be an even integer. In this short note we prove that the Jacobian variety of a quasiplatonic Riemann surface with associated group of automorphisms isomorphic to C 2 2 ⋊2 Cm admits complex multiplication. We then extend this result to provide a criterion under which the Jacobian variety of a quasiplatonic Riemann surface admits complex multiplication.

Research paper thumbnail of Groups of automorphisms of Riemann surfaces and maps of genus p + 1 where p is prime

Annales Fennici Mathematici

We classify compact Riemann surfaces of genus g, where g − 1 is a prime p, which have a group of ... more We classify compact Riemann surfaces of genus g, where g − 1 is a prime p, which have a group of automorphisms of order ρ(g − 1) for some integer ρ ≥ 1, and determine isogeny decompositions of the corresponding Jacobian varieties. This extends results of Belolipetzky and the second author for ρ > 6, and of the first and third authors for ρ = 3, 4, 5 and 6. As a corollary we classify the orientably regular hypermaps (including maps) of genus p + 1, together with the non-orientable regular hypermaps of characteristic −p, with automorphism group of order divisible by the prime p; this extends results of Conder,Širáň and Tucker for maps.

Research paper thumbnail of Riemann surfaces of genus 1 + q2 with 3q2 automorphisms

Research paper thumbnail of A generalisation of Kani-Rosen decomposition theorem for Jacobian varieties

ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE

In this short paper we generalise a theorem due to Kani and Rosen on decomposition of Jacobian va... more In this short paper we generalise a theorem due to Kani and Rosen on decomposition of Jacobian varieties of Riemann surfaces with group action. This generalisation extends the set of Jacobians for which it is possible to obtain an isogeny decomposition where all the factors are Jacobians.

Research paper thumbnail of On families of Riemann surfaces with automorphisms

Journal of Pure and Applied Algebra

Research paper thumbnail of On Riemann surfaces of genus g with 4g–4 automorphisms

Israel Journal of Mathematics

In this article we study compact Riemann surfaces with a nonlarge group of automorphisms of maxim... more In this article we study compact Riemann surfaces with a nonlarge group of automorphisms of maximal order; namely, compact Riemann surfaces of genus g with a group of automorphisms of order 4g − 4. Under the assumption that g − 1 is prime, we provide a complete classification of them and determine isogeny decompositions of the corresponding Jacobian varieties.

Research paper thumbnail of On Jacobians with group action and coverings

Mathematische Zeitschrift

Let S be a compact Riemann surface and let H be a finite group. It is known that if H acts on S t... more Let S be a compact Riemann surface and let H be a finite group. It is known that if H acts on S then there is a H-equivariant isogeny decomposition of the Jacobian variety JS of S, called the group algebra decomposition of JS with respect to H. If S 1 → S 2 is a regular covering map, then it is also known that the group algebra decomposition of JS 1 induces an isogeny decomposition of JS 2. In this article we deal with the converse situation. More precisely, we prove that the group algebra decomposition can be lifted under regular covering maps, under appropriate conditions.

Research paper thumbnail of On the Connectedness of the Singular Locus of the Moduli Space of Principally Polarized Abelian Varieties

Moscow Mathematical Journal

Let Ag denote the moduli space of principally polarized abelian varieties of dimension g ≥ 3. In ... more Let Ag denote the moduli space of principally polarized abelian varieties of dimension g ≥ 3. In this paper we prove the connectedness of the singular sublocus of Ag consisting of those abelian varieties which possess an involution different from −id.

Research paper thumbnail of On the one-dimensional family of Riemann surfaces of genus q with 4q automorphisms

Journal of Pure and Applied Algebra

Bujalance, Costa and Izquierdo have recently proved that all those Riemann surfaces of genus g ≥ ... more Bujalance, Costa and Izquierdo have recently proved that all those Riemann surfaces of genus g ≥ 2 different from 3, 6, 12, 15 and 30, with exactly 4g automorphisms form an equisymmetric one-dimensional family, denoted by Fg. In this paper, for every prime number q ≥ 5, we explore further properties of each Riemann surface S in Fq as well as of its Jacobian variety JS.

Research paper thumbnail of A family of (p, n)-gonal Riemann surfaces with several (p, n)-gonal groups

arXiv: Algebraic Geometry, 2020

Let pgeqslant3p \geqslant 3pgeqslant3 be a prime number and let ngeqslant0n \geqslant 0ngeqslant0 be an integer such that p−1p-1p−1 divid... more Let pgeqslant3p \geqslant 3pgeqslant3 be a prime number and let ngeqslant0n \geqslant 0ngeqslant0 be an integer such that p−1p-1p−1 divides n.n.n. In this short note we construct a family of (p,n)(p,n)(p,n)-gonal Riemann surfaces of maximal genus 2np+(p−1)22np+(p-1)^22np+(p−1)2 with more than one (p,n)(p,n)(p,n)-gonal group.

Research paper thumbnail of Weil's Galois Descent Theorem from a computational point of view

Let L/ K be a finite Galois extension and let X be an affine algebraic variety defined over L. We... more Let L/ K be a finite Galois extension and let X be an affine algebraic variety defined over L. Weil's Galois descent theorem provides necessary and sufficient conditions for X to be definable over K, that is, for the existence of an algebraic variety Y defined over K together with a birational isomorphism R:X → Y defined over L. Weil's proof does not provide a method to construct the birational isomorphism R. The aim of this paper is to give an explicit construction of R.

Research paper thumbnail of Nilpotent groups of automorphisms of families of Riemann surfaces

In this article we extend results of Zomorrodian to determine upper bounds for the order of a nil... more In this article we extend results of Zomorrodian to determine upper bounds for the order of a nilpotent group of automorphisms of a complex d-dimensional family of compact Riemann surfaces, where d ≥ 1. We provide conditions under which these bounds are sharp and, in addition, for the one-dimensional case we construct and describe an explicit family attaining the bound for infinitely many genera. We obtain similar results for the case of p-groups of automorphisms.

Research paper thumbnail of Weil’s Galois descent theorem from a computational point of view

arXiv: Algebraic Geometry, 2012

Let mathcalL/mathcalK{\mathcal L}/{\mathcal K}mathcalL/mathcalK be a finite Galois extension and let XXX be an affine algebraic v... more Let mathcalL/mathcalK{\mathcal L}/{\mathcal K}mathcalL/mathcalK be a finite Galois extension and let XXX be an affine algebraic variety defined over mathcalL{\mathcal L}mathcalL. Weil's Galois descent theorem provides necessary and sufficient conditions for XXX to be definable over mathcalK{\mathcal K}mathcalK, that is, for the existence of an algebraic variety YYY defined over mathcalK{\mathcal K}mathcalK together with a birational isomorphism R:XtoYR:X \to YR:XtoY defined over mathcalL{\mathcal L}mathcalL. Weil's proof does not provide a method to construct the birational isomorphism R.R.R. The aim of this paper is to give an explicit construction of RRR.

Research paper thumbnail of On the fiber product of Riemann surfaces II

Let S0,S1S_{0}, S_{1}S0,S1 and S2S_{2}S2 be connected Riemann surfaces and let beta1:S1toS0\beta_{1}:S_{1} \to S_{0}beta1:S1toS0 ... more Let S0,S1S_{0}, S_{1}S0,S1 and S2S_{2}S2 be connected Riemann surfaces and let beta1:S1toS0\beta_{1}:S_{1} \to S_{0}beta1:S1toS0 and beta2:S2toS0\beta_{2}:S_{2} \to S_{0}beta2:S2toS0 be surjective holomorphic maps. The associated fiber product S1times(beta1,beta2)S2S_{1} \times_{(\beta_{1},\beta_{2})} S_{2}S1times(beta1,beta2)S2 has the structure of a singular Riemann surface, endowed with a canonical map beta\betabeta to S0S_{0}S0 satisfying that betajcircpij=beta\beta_{j} \circ \pi_{j}=\betabetajcircpij=beta, where pij\pi_{j}pij is coordinate projection onto SjS_{j}Sj. In this paper we provide a Fuchsian description of the fiber product and obtain that if one the maps betaj\beta_{j}betaj is a regular branched cover, then all its irreducible components are isomorphic. In the case that both betaj\beta_{j}betaj are of finite degree, we observe that the number of irreducible components is bounded above by the greatest common divisor of the two degrees; we study the irreducibility of the fiber product. In the case that S0=widehatmathbbCS_{0}=\widehat{\mathbb C}S0=widehatmathbbC, and S1S_{1}S1 and S2S_{2}S2 are compact, we define the strong field of moduli of the pair $(S_{1} ...

Research paper thumbnail of A note on Jacobians of quasiplatonic Riemann surfaces with complex multiplication

Geometriae Dedicata

Let m 6 be an even integer. In this short note we prove that the Jacobian variety of a quasiplato... more Let m 6 be an even integer. In this short note we prove that the Jacobian variety of a quasiplatonic Riemann surface with associated group of automorphisms isomorphic to C 2 2 ⋊2 Cm admits complex multiplication. We then extend this result to provide a criterion under which the Jacobian variety of a quasiplatonic Riemann surface admits complex multiplication.

Research paper thumbnail of Groups of automorphisms of Riemann surfaces and maps of genus p + 1 where p is prime

Annales Fennici Mathematici

We classify compact Riemann surfaces of genus g, where g − 1 is a prime p, which have a group of ... more We classify compact Riemann surfaces of genus g, where g − 1 is a prime p, which have a group of automorphisms of order ρ(g − 1) for some integer ρ ≥ 1, and determine isogeny decompositions of the corresponding Jacobian varieties. This extends results of Belolipetzky and the second author for ρ > 6, and of the first and third authors for ρ = 3, 4, 5 and 6. As a corollary we classify the orientably regular hypermaps (including maps) of genus p + 1, together with the non-orientable regular hypermaps of characteristic −p, with automorphism group of order divisible by the prime p; this extends results of Conder,Širáň and Tucker for maps.

Research paper thumbnail of Riemann surfaces of genus 1 + q2 with 3q2 automorphisms

Research paper thumbnail of A generalisation of Kani-Rosen decomposition theorem for Jacobian varieties

ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE

In this short paper we generalise a theorem due to Kani and Rosen on decomposition of Jacobian va... more In this short paper we generalise a theorem due to Kani and Rosen on decomposition of Jacobian varieties of Riemann surfaces with group action. This generalisation extends the set of Jacobians for which it is possible to obtain an isogeny decomposition where all the factors are Jacobians.

Research paper thumbnail of On families of Riemann surfaces with automorphisms

Journal of Pure and Applied Algebra

Research paper thumbnail of On Riemann surfaces of genus g with 4g–4 automorphisms

Israel Journal of Mathematics

In this article we study compact Riemann surfaces with a nonlarge group of automorphisms of maxim... more In this article we study compact Riemann surfaces with a nonlarge group of automorphisms of maximal order; namely, compact Riemann surfaces of genus g with a group of automorphisms of order 4g − 4. Under the assumption that g − 1 is prime, we provide a complete classification of them and determine isogeny decompositions of the corresponding Jacobian varieties.

Research paper thumbnail of On Jacobians with group action and coverings

Mathematische Zeitschrift

Let S be a compact Riemann surface and let H be a finite group. It is known that if H acts on S t... more Let S be a compact Riemann surface and let H be a finite group. It is known that if H acts on S then there is a H-equivariant isogeny decomposition of the Jacobian variety JS of S, called the group algebra decomposition of JS with respect to H. If S 1 → S 2 is a regular covering map, then it is also known that the group algebra decomposition of JS 1 induces an isogeny decomposition of JS 2. In this article we deal with the converse situation. More precisely, we prove that the group algebra decomposition can be lifted under regular covering maps, under appropriate conditions.

Research paper thumbnail of On the Connectedness of the Singular Locus of the Moduli Space of Principally Polarized Abelian Varieties

Moscow Mathematical Journal

Let Ag denote the moduli space of principally polarized abelian varieties of dimension g ≥ 3. In ... more Let Ag denote the moduli space of principally polarized abelian varieties of dimension g ≥ 3. In this paper we prove the connectedness of the singular sublocus of Ag consisting of those abelian varieties which possess an involution different from −id.

Research paper thumbnail of On the one-dimensional family of Riemann surfaces of genus q with 4q automorphisms

Journal of Pure and Applied Algebra

Bujalance, Costa and Izquierdo have recently proved that all those Riemann surfaces of genus g ≥ ... more Bujalance, Costa and Izquierdo have recently proved that all those Riemann surfaces of genus g ≥ 2 different from 3, 6, 12, 15 and 30, with exactly 4g automorphisms form an equisymmetric one-dimensional family, denoted by Fg. In this paper, for every prime number q ≥ 5, we explore further properties of each Riemann surface S in Fq as well as of its Jacobian variety JS.

Research paper thumbnail of A family of (p, n)-gonal Riemann surfaces with several (p, n)-gonal groups

arXiv: Algebraic Geometry, 2020

Let pgeqslant3p \geqslant 3pgeqslant3 be a prime number and let ngeqslant0n \geqslant 0ngeqslant0 be an integer such that p−1p-1p−1 divid... more Let pgeqslant3p \geqslant 3pgeqslant3 be a prime number and let ngeqslant0n \geqslant 0ngeqslant0 be an integer such that p−1p-1p−1 divides n.n.n. In this short note we construct a family of (p,n)(p,n)(p,n)-gonal Riemann surfaces of maximal genus 2np+(p−1)22np+(p-1)^22np+(p−1)2 with more than one (p,n)(p,n)(p,n)-gonal group.