Here are a few project ideas:

Elliptic curves (characterize them, algebraic structure, analytic properties,etc.)

- Source: Koblitz: Introduction to Elliptic Curves and Modular Forms

Blowing-up (plane curve) singularities

- Source: Harris: A First Course in Algebraic Geometry

Newton polygons and Puiseux expansions

- Source: Brieskorn, Knorrer: Plane Algebraic Curves

Genus of a curve, or other topological properties

- Source: Miranda, Algebraic Curves and Riemann Surfaces

Kinematics problem for robotics

- Source: Cox, Little, O'Shea: Ideals, Varieties, and Algorithms

Automatic geometric theorem proving

- Source: Cox, Little, O'Shea: Ideals, Varieties, and Algorithms

Improvements to Buchberger's algorithm, other algorithmic methods

- Source: Cox, Little, O'Shea: Ideals, Varieties, and Algorithms;

Greuel, Pfister, Bachmann: A Singular Introduction to Commutative Algebra

- Source: Cox, Little, O'Shea: Ideals, Varieties, and Algorithms;
Investigate a distinguished curve

- Source: Famous Curves Index

Please talk with me about the plan for your project to get some ideas or sources, or if you want some other project ideas.