110 research outputs found
On Hyperfocused Arcs in PG(2,q)
A k-arc in a Dearguesian projective plane whose secants meet some external
line in k-1 points is said to be hyperfocused. Hyperfocused arcs are
investigated in connection with a secret sharing scheme based on geometry due
to Simmons. In this paper it is shown that point orbits under suitable groups
of elations are hyperfocused arcs with the significant property of being
contained neither in a hyperoval, nor in a proper subplane. Also, the concept
of generalized hyperfocused arc, i.e. an arc whose secants admit a blocking set
of minimum size, is introduced: a construction method is provided, together
with the classification for size up to 10
Algebraic curves with many automorphisms
Let be a (projective, geometrically irreducible, nonsingular) algebraic
curve of genus defined over an algebraically closed field of odd
characteristic . Let be the group of all automorphisms of which
fix element-wise. It is known that if then the -rank
(equivalently, the Hasse-Witt invariant) of is zero. This raises the
problem of determining the (minimum-value) function such that whenever
then has zero -rank. For {\em{even}} we prove
that . The {\em{odd}} genus case appears to be much more
difficult although, for any genus , if has a solvable
subgroup such that then has zero -rank and fixes a
point of . Our proofs use the Hurwitz genus formula and the Deuring
Shafarevich formula together with a few deep results from finite group theory
characterizing finite simple groups whose Sylow -subgroups have a cyclic
subgroup of index . We also point out some connections with the Abhyankar
conjecture and the Katz-Gabber covers
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