we introduce the basic definitions and properties of scheme and cohomology then applythe language of scheme and cohomology to describe the curves from genus 0 to genus 6. Firstly,we give the basic definitions of scheme and some properties which we usually use. Secondly, wedescribe the process of computing the cohomology group of projective space which is enough formost cases. We also introduce divisor and invertible sheaf which are essential for the followingstud of curves. Finally we apply the techniques developed above to study curves, focusing oncurves from genus 0 to genus 6.