#### Alladi Ramakrishnan Hall

#### A Representation Theorem of Line Arrangements and its Generalization to Hyperplane Arrangements via Convex Positive Bijections

#### C P Anil Kumar

##### Center for Science,Technology and Policy (CSTEP) Bengaluru

*In this talk we first show that any line arrangement over a field with 1-ad structure can be isomorphically represented by a set of lines of same cardinality with a given set of distinct slopes. Then we generalize this theorem to higher dimensional hyperplane arrangements over a field with 1-ad structure. We prove, using a certain observation on the theme of central points, that, any two hyperplane arrangements are isomorphic modulo translations of any hyperplane if and only if there is a convex positive bijection between the corresponding associated normal systems. Finally we exhibit two normal systems in three dimensions of cardinality six which are not isomorphic.*

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