Several foundational notions from Algebraic Geometry

I will not cover anything related to Lorentzian polynomials this time. Instead, I plan to introduce several foundational notions from algebraic geometry, including:

• Projective space
• Projective varieties
• Weil divisors
• Cartier divisors
• Effective Cartier divisors
• Divisor class and Picard group
• Smoothness
• \(\mathbb{Q}-divisors\)

Some of these topics are not necessary for our main goal. After talking with Dima today, it seems that many of these concepts are best understood from a geometric perspective rather than only through formal algebraic definitions. So this session will focus more on building geometric intuition.

If possible, please review the definitions of projective space (as \(Proj (R[x_1, x_2, \dots, x_n]))\) and projective varieties (an integral closed subscheme of a projective space).