Arc Schemes in Logarithmic Algebraic Geometry

Abstract

We develop the theory of log arc schemes of algebraic varieties, building on prior work by Noguchi, Vojta, Dutter, Karu and Staal, and others on log jet schemes. These are analogues to the ordinary arc and jet schemes of a variety, in the category of log schemes in the sense of Kato. In particular we characterise the fine log schemes for which the associated log arc scheme is irreducible, generalising a well-known theorem of Kolchin to the log geometry setting, and develop a theory of integration on log arc schemes generalising the theory of motivic integration on ordinary arc schemes due to Kontsevich, Denef and Loeser, and Batyrev. Prior to this we give an essentially self-contained exposition of the elements of log geometry that we require.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/111507/1/bkflem_1.pd