SOME CASES OF OORT'S CONJECTURE ABOUT NEWTON POLYGONS OF CURVES

This paper contains a method to prove the existence of smoothcurves in positive characteristic whose Jacobians have unusual Newton poly-gons. Using this method, I give a new proof that there exist supersingular curvesof genus 4 in every prime characteristic. More generally, the main result of thepaper is that, for everyg >= 4 and primep, every Newton polygon whosep-rankis at leastg-4 occurs for a smooth curve of genusgin characteristicp.Inaddition, this method resolves some cases of Oort's conjecture about Newtonpolygons of curves