1. ALGEBRAIC CURVES

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A plane non-singular algebraic curve of degree n is called maximally symmetric if its group of automorphisms meets the corresponding bound over the complex numbers with equality. The objective of a series of papers was an attempt to classify the maximally symmetric plane non-singular algebraic curves of arbitrary degree defined over an algebraically closed field of characteristic zero.
As a starting point particular values of n were studied: in [13] the cases n<20, n prime, were considered. In all those cases the most symmetric curves are projectively equivalent to the Fermat curve of equation x^n + y^n + z^n  =0 whose group of automorphisms has order 6n^2.
As a next step I considered the first non-prime degree n=6 which had not been completely classified (for n=4 one has the well-known Klein quartic). It is proved in [29] that the Wiman sextic is the unique maximally symmetric curve in this case. It is invariant under the alternating group A_6. This generalizes a result by H. Doi, K. Idei and H. Kaneta which is valid only over the complex numbers. Moreover in [29] it has been defined an immersion of PSL(2,7) and an immersion of A_6 in PGL(3,k), which allow to determine family of arcs with big automorphism group, projectively equivalent to the flexes of the Klein quartic or the Wiman's sextic, as described in the following paragraph. It should be noted that the first immersion has been used by G. Korchmaros ad  L. Indaco (2011) and the second in [55] to determine infinite family of arcs with large automorphism group. The next step has been to generalize the result of [13] to every prime degree. In [AC20] and [61] I obtained a more general result: for every n different from 4 and 6, up to projectivity the Fermat curve is the unique non-singular algebraic plane curve which is maximally symmetric. This generalization requested a preliminary study of group theory in order to determine general properties for subgroups of PGL(3,k) such that the studied curves are invariant with respect to them.

Other types of works, concerning the connections between Finite Geometry and Algebraic curve theory are illustrated in  next section Galois Geometries.