Daily Assignment for Monday, January 27

Due Wednesday, January 29 on Canvas (if you want me to look at your work).

• Finish reading chapter 4 of Gathmann's book on curves if you haven't already.
• You may want to look back at Proposition 2.26 in chapter 2.
• Fill in the details from our proof of the 2-variable Nullstellensatz from class. The main point that was missing was to show that if \( \ell \) is a line and \( \ell \cap F = \varnothing \) then \( \ell \) is an asymptote of \( F \). You could do this by proving Bézout's theorem for intersections of curves and lines.
• Can you demonstrate the proposition from class : if \( F \) and \( G \) are homogeneous polynomials in \( K[x,y,z] \) such that \( F \cap G \cap z = \varnothing \) then the dehomoginization homomorphism \( K[x,y,z]_d / (F,G)_d \to K[x,y] / (F^i, G^i) \) is an isomorphism for \( d \gg 0 \) ? Suggestion : prove it is surjective and injective. Surjectivity should be easier.
• Finish the proof of Bézout's theorem from class by computing \( \dim_K (F)_d \), \( \dim_K (G)_d \), \( \dim_K (FG)_d \), and \( \dim_K K[x,y,z]_d \) and combining them to get \( \dim_K K[x,y,z]_d / (F,G)_d \).