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We study the 2Weierstrass points on quartic curves. If the curve has a cyclic covering structure over P(C), then the computation of 2Weierstrass points is relatively easy (see Chapter 3). We deal with a 1parameter family of smooth quartic curves without cyclic covering structures over P(C). Let Ca be the smooth plane quartic defined by the equation: F(x,y,z)=x4+y4+z4+a(xy+xz+yz)=0, a?1,2. The 1Weierstrass points on Ca were extensively studied by Kuribayashi and his students, around 1980s. We call these quartic curves Kuribayashi quartics. In this book, we give the geometric classification of the 2Weierstrass points on Kuribayashi quartics (see Chapter 2). In chapter 4, we study the 1Weierstrass points on quintic curves, we see that a 1Weierstrass point P of a smooth plane quintic C is either a flex or a sextactic point. Finally, we compute the 1Weierstrass points on two 1parameter families of singular plane quintics by computing special adjoint conics at these points.
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