Perhaps you're going about this the wrong way. Instead of trying to describe what most algebraic geometers do today, try to describe a problem, or set of problems, that is reasonably concrete and accessible, and go from there. I think there are plenty of topics to choose from. You've already mentioned solving systems of equations. Even if most algebraic geometers don't try to find solutions, they certainly want to know when solutions exists (Nullstellensatz) and how many parameters are required (dimension). Another historically important motivation is the study ellipitic/abelian integrals: you can go from addition laws for integrals to additional laws on the curve/Jacobian...

Perhaps you're going about this the wrong way. Instead of trying to describe what most algebraic geometers do today, try to describe a problem, or set of problems, that is reasonably concrete and accessible, and go from there. I think there are plenty of topics to choose from. You've already mentioned solving systems of equations. Even if most algebraic geometers don't try to find solutions, they certainly want to know when solutions exist (Nullstellensatz) and how many parameters are required (dimension). Another historically important motivation is the study of elliptic/abelian integrals: you can go from addition laws for integrals to addition laws on the curve/Jacobian...

Perhaps you're going about this the wrong way. Instead of trying to describe what most algebraic geometers do today, try to describe a problem, or set of problems, that is reasonably concrete and accessible, and go from there. I think there are plenty of topics to choose from. You've already mentioned solving systems of equations. Another historically important motivation is the study ellipitic/abelian integrals: you can go from addition laws for integrals to additional laws on the curve/Jacobian...

Perhaps you're going about this the wrong way. Instead of trying to describe what most algebraic geometers do today, try to describe a problem, or set of problems, that is reasonably concrete and accessible, and go from there. I think there are plenty of topics to choose from. You've already mentioned solving systems of equations. Even if most algebraic geometers don't try to find solutions, they certainly want to know when solutions exists (Nullstellensatz) and how many parameters are required (dimension). Another historically important motivation is the study ellipitic/abelian integrals: you can go from addition laws for integrals to additional laws on the curve/Jacobian...