Classification Of Regular Projective Curves Of Genus One Over An Imperfect Field

Regular algebraic curves over an algebraically closed field are all smooth,and their deformation equivalence classes are determined by their genus.Regular curves over a non-algebraically closed field of characteristic 0 can be reduced to an algebraically closed field,but regular curves over an imperfect field of characteristic p may not be smooth,and new phenomena occur when reducing them to an algebraically closed field.The case where the arithmetic genus is 0 is clear.In this paper,we will study non-smooth regular curves of arithmetic genus 1 over an imperfect field of characteristic p under base change.And this situation only occurs when the characteristic p is 2,3.On the one hand,we will use the Riemann-Roch theorem to study the properties of the linear system on the curves of the arithmetic genus 1.On the other hand,we try to use geometric methods to construct corresponding examples for various possible phenomena.