Algebraic Geometry Jottings

This is the index page for the series of posts, Algebraic Geometry Jottings.

Texts: Fulton’s Algebraic Curves and Kendig’s A Guide to Plane Algebraic Curves.

AGJ-1: Bézout’s Theorem. Newton’s anticipation. The two curves (from Fulton) in our running example. How points can “go missing”.

AGJ-2: Proper statement of Bézout’s Theorem. Imaginary points. Points at infinity in the projective plane. Changing the “viewing window” to make these show up in diagrams. A bit of art history.

AGJ-3: Multiplicity: informal “perturbation” definition, leading to Kendig’s first definition.

AGJ-4: Multiplicity between two branches. First glimmers of the resultant. Kendig’s second definition.

AGJ-5: Fulton’s seven properties that characterize multiplicity, aka intersection number. Sample computation, for the two roses.

AGJ-6: A bare minimum of commutative algebra: ideals, homomorphisms, PIDs and UFDs.

AGJ-7: Fulton’s second definition. The coordinate ring and the local ring. Very simple example: a parabola and a horizontal line.

AGJ-8: A “chewing” post: we mull over the material of the previous post.

AGJ-9: The Resultant, Episode 1: The framework, and the “double-product” formula for the resultant.

AGJ-10: The Resultant, Episode 1, Inside the Episode: Another “chewing” post. The two-ellipses example. Identically zero vs. zero at a point.

AGJ-11: The Resultant, Episode 2: As above, so below: R⊆K⊆L; “coprime in R” iff “coprime in L“. The linear map Φ(p,q)=pE+qF, and its matrix, the Sylvester matrix.

AGJ-12: The Resultant, Episode 3: More about the Sylvester matrix and its determinant. The equation PE+QF=det(S).

AGJ-13: The Resultant, Episode 3, Inside the Episode: A “chewing” post. Key facts about the roots of the resultant, its order, and its degree. Examples: two roses; two hyperbolas; two more hyperbolas.

AGJ-14: The Resultant, Episode 4: Proof that det(S) equals the double product.

AGJ-15: The Resultant, Episode 5 (The Finale): Recap of the framework and the key formulas and facts. Proofs of Facts 1 to 3.

AGJ-16: The Resultant, Episode 5, Inside the Episode: A “near proof” of Fact 3. The case of zero leading coefficients. A variation on the proof in Episode 4. The formula PE+QF=D revisited.

AGJ-17: Starting Kendig’s proof of Bézout’s Theorem: outline; proof of Facts 4 and 5, with a special assumption.

AGJ-18: Fractional power series, and using them to eliminate the special assumption.