That last book devolved a little bit too much into methods for my taste, and so I was missing my algebra stuff. I found a book that seemed like it dealt a little bit with more geometry-adjacent stuff, which is a topic I'm constantly trying to find better access to. Complex Algebraic Surfaces, Second Edition by Arnaud Beauville, published through Cambridge University Press, seems like it's going to bridge the gap between the algebraic, topology topics I like, and the geometry stuff I tend to find difficult. Since we're working with a new book, I would like to create slightly more work for me. Originally, I did some of the questions in the books, and I want to return to that, now that I'm not actively studying at a university anymore, I'd like to keep my skills sharp and that's probably the best way to do it. I don't think I'll include the stuff in these weekly study notes, because I don't actually want to give myself weekly homework, and sometimes the chunks I work through are too short to get to the problems; and sometimes the books don't feature practice problems at all. I'll just make note when I decide to use a problem from a book.
This book starts off with a lot of topology presupposed. It assumes, by default, surfaces with Zariski topology, and it will assume sheafs are coherent algebraic sheafs (p. 1). The Picard group Pic S is defined on a smooth variety S and contains the isomorphism classes of invertible sheaves (or line bundles) on S. Every effective divisor D on S has a corresponding invertible sheaf OS(D) and a section s ∈ H0(OS(D)), s &neq; 0 unique up to scalar multiplication s.t. div(s) = D. The sheaf is arbitrary by linearity and the linear map D → OS(D) identifies Pic S with the group of linear equivalence classes of divisors on S. Given X a smooth variety and the morphism f: S → X, the inverse image to f of an invertible sheaf is defined to a homomorphism f*: Pic X → Pic S. If f is surjective, then the inverse image of a divisor exists s.t. f*OX(D) = OS(f*(D)). If f is a morphism of surfaces with generically finite degree d, the direct image f*C of an irreducible curve C is 0 if, f(C) is a point and rΓ, if it's a curve Γ with and the morphism C → Γ induced by f having finite degree r. f*. f*f*D = dD ∀ D on S (I.1). Two distinct irreducible curves C, C' on a surface S, and x ∈ C ∩ C', Ox the local ring of S at x. If f (g) is an equation of C (C') in Ox, the intersection multiplicity of C, C' at x is mx(C∩C') = dimℂOx/(f, g) (I.2). Given C, C' distinct on S, the intersection number (C.C') = ∑x ∈ C∩C' mx(C∩C') (I.3). ∀ L, L' ∈ Pic S: (L.L') = χ(OS) - χ(L-1) - χ(L'-1) + χ(L-1⊗L'-1) ⇒ ( . ) is a symmetric bilinear form on Pic S, and (OS(C).OS(C')) = (C.C') (I.4). A divisor D on S and a hyperplane section H of S &exists; n ≥ 0 : D + nH a hyperplane section (for another embedding). Write D ≡ A - B where A and B are smooth curves on S, A ≡ D + nH, B ≡ nH (I.7). In application, a smooth curve C with f: S → C surjective and a fibre of f implies F2 = 0. If a S' is a surface and g: S → S' a generically finite morphism of degree d; D, D' divisors on S; then g*D.g*D' = d(D.D') (I.8). For analytic sheaves on S and ordinary topology, the sheaf of holomorphic functions on S hOS is considered an analytic manifold. The exponential map e: hOS → hO*S; e(f) = e2πif that is locally surjective with kernel that consists of locally constant integer-valued functions. There is an exact sequence 0 → ℤ → hOS →e hO*S → 1 (I.10). A surface S with line bundle L and ωS the line bundle of differential 2-forms on S, then is H2(S, ωS) is a 1D vector space. This defines the duality Hi(S, L) ⊗ H2-i(S, ωS ⊗ L-1) → H2(S, ωS) → ℂ. Also, χ(L) = χ(ωSL-1) (I.11). ∀ L ∈ Pic S: χ(L) = χ(OS) + 1/2 (L2 - L.ωS) (Riemann-Roch, I.12). This is used to derive Noether's formula (I.14) and the genus formula (I.15).