The article “On algebraic curves with many automorphisms in characteristic p” by the algebra group member Maria Montanucci has been accepted for publication in Mathematische Zeitschrift.
In this article, Maria gives a partial answer to an open problem regarding the size and the action of large automorphism groups of agebraic curves in positive characteristic. A preprint of this article can be found here
A preprint on generalized Weierstrass semigroups on a family of maximal curves is now available on arXiv here.
It was written by algebra group member Maria Montanucci and Guilherme Tizziotti from Universidade Federal de Uberlândia (UFU), Brazil. In the preprint the generalized Weiestrass semigroup at several points of a family of maximal curves which is not covered by the Hermitian curve is computed.
The paper “A class of linear sets in PG(1,q^5)” by Maria and Corrado Zanella appeared online in the journal Finite Fields and its Applications. In the manuscript Maria and Corrado study some interesting combinatorial structures called maximum scattered linear sets over projective spaces. These structures have been intensively studied during the last year, particularly for the connection to coding theory (MRD codes).
Maximum scattered linear sets over PG(1,q^n) have been completely classified for n at most 4 by Csajbók-Zanella (2018) and Lavrauw-Van de Voorde (2010). In this paper Maria and Corrado analyze the case n=5. There a wide class of linear sets is studied which depends on two parameters. Conditions for the existence, in this class, of possible new maximum scattered linear sets in are exhibited.
The paper “An Fp2-maximal Wiman’s sextic and its automorphisms” by Maria, Massimo Giulietti, Motoko Kawakita and Stefano Lia appeared online in the journal Advances in Geometry. In this paper Maria, Massimo, Motoko and Stefano study a sextic curve introduced in 1895 by Wiman as a Riemann surface over the complex field, but seen over finite fields. They showed that its full automorphism group is isomorphic to the symmetric group S5, generalizing Wiman’s result. It is also shown that when the finite field has cardinality 19^2 then the Wiman’s sextic is maximal and it is not Galois covered by the Hermitian curve.
Maria has accepted to become editor of the Elsevier journal “Journal of Combinatorial Theory Series A” for the next 3 years. The Journal of Combinatorial Theory, Series A is one of the premier journals on theoretical and practical aspects of combinatorics in all branches of science. The journal is primarily concerned with finite and discrete structures, designs, finite geometries, codes, combinatorics with number theory, combinatorial games, extremal combinatorics, combinatorics of storage, and other important theory/applications of combinatorics.
Today Maria’s paper “On a conjecture about maximum scattered subspaces of Fq6×Fq6” appeared online in Linear Algebra and its Applications, as a joint work with Daniele Bartoli and Bence Csajbók. Here Bence, Daniele and Maria proved a conjecture from 2018 regarding a family of maximum scattered subspaces of Fq6 x Fq6. A preprint is available here.
Today, Maria gave a talk for the online conference SIAM Conference on Applied Algebraic Geometry (AG21). Her talk, titled “Error correcting codes from maximal curves”, was a part of the mini-symposium Polynomial and Evaluation Codes and their Applications. In her talk Maria presented several construction of error correcting codes (AG codes, quantum codes and locally recoverable codes) using the nice structure of automorphism groups of maximal curves.
Today, Maria’s paper “Fp2-maximal curves with many automorphisms are Galois-covered by the Hermitian curve” appeared in the journal Advances in Geometry, as a joint work with Daniele Bartoli and Fernando Torres. Here a open problem dated 2000 is analyzed. Namely, is it true that every Fp2-maximal curve (where p is a prime) is covered by the Hermitian curve? In this paper Maria, Daniele and Fernando show that this is the case provided that the curve has a sufficiently large automorphism group. Also, the first example of an Fp2-maximal curve which is not Galois-covered by the Hermitian curve is presented. A preprint is available here.
Peter, Maria and Leonardo’s paper “Weierstrass semigroups on the Skabelund maximal curve” has been accepted for publication in the journal Finite Fields and their Applications. Here the Weierstrass semigroup is computed at every point of the Skabelund maximal curve (covering the famous Suzuki curve), as well as the Weierstrass points of the curve. Since these are the main ingredients to construct AG codes, the result has interesting consequence in applications. Also, this is one of the few examples in the literature in which this complete analysis is known. A preprint is available here.
Today Maria’s paper “On certain self-orthogonal AG codes with applications to Quantum error-correcting codes” appeared in the journal Designs, Codes and Cryptography, as a joint work with Daniele Bartoli and Giovanni Zini. Here, a construction of quantum codes from self-orthogonal algebraic geometry codes is provided as well as on a new family of algebraic curves to which the construction is applied, named Swiss curves. The reason behind this curious name, is that the authors developed the key ideas proposed in this publication during the conference SIAM AG19, which took place in Bern (Switzerland). A preprint is available here.