Single-element Extensions Of Hyperplane Arrangements

In 1965,Crapo introduced the single-element extension of a matroid and classified all single-element extensions.In 1978,Las Vergnas extended Crapo’s classification to oriented matroids.In 2022,Fu-Wang studied the isomorphic classification on all single-element extensions of a represented matroid,and characterized all isomorphism classes via its adjoint matroid.Note that each single-element extension of a represented matroid can be identified with a linear single-element extension of the corresponding linear hyperplane arrangement.In this thesis,we will study single-element extensions of a general hyperplane arrangement,including the linear and affine hyperplane arrangements.Listed below are our main results.Firstly,we extend the definition of the adjoint matroid and define an adjoint arrangement for a general hyperplane arrangement.Applying the intersection lattice of the adjoint arrangement,we characterize all isomorphism classes of single-element extensions of a hyperplane arrangement.Secondly,with above classification we will continue to study several combinatorial invariants of each class,including the characteristic polynomial,the Whitney polynomial,two kinds of Whitney numbers,the number of faces,and the number of regions.We show that these invariants are upper semi-continuous on the intersection lattice of the adjoint arrangement.In the end,with the deletion and restriction of hyperplane arrangements,we are capable of studying the classification on all restrictions of a hyperplane arrangement and there invariants.