Results on the singular surface and their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Regular points — Proposition 1.4. At a regular point of a finitely generated algebra over an algebraically closed field, the derivation module is free of rank equal to the dimension, also after a principal localization.
• A counterexample in characteristic p — Proposition 2.1. k[x,y,z]/(xy - z^p) in characteristic p is a normal domain of dimension two, singular at the origin, with free derivation module on x d/dx - y d/dy and d/dz.
• Every dimension — Proposition 3.4. Adjoining variables to a two-dimensional counterexample gives normal domains of every dimension n+2 with free derivation module and nonregular locus of codimension two.
• Adjunction on the quadric — Lemma 1.3. For a smooth curve of bidegree (c,d) on P^1 x P^1, the residue gives an isomorphism of O(c-2,d-2) restricted to the curve with its canonical bundle, with explicit chart formulas.
• Switches and torsion — Lemma 2.3. On a smooth (2,2) curve, the composite T of the two switches satisfies O(T(p) - p) = S - F, and k(S - F) is trivial exactly when T^k(p) = p.
• An explicit elliptic curve with a point of order four — Proposition 2.4. The smooth (2,2) curve through the eight-point zigzag on the grid {0,1,2,3}^2 has S - F of exact order four.
• The quadratic form — Proposition 3.3. A unique element of Sym^2 of the sections of Phi multiplies to -eta/omega; its quadratic form is nondegenerate, isotropic exactly at the ramification points of the first projection.
• Six vectors — Lemma 4.1. In a nondegenerate quadratic plane: six nonzero vectors in three distinct nonisotropic directions, avoiding a finite set, with w^2 = 90 and w_j^2 + w_j.w = 1 for their sum w.
• A curve of genus 21 with fourth-order contact — Proposition 1.4. A smooth curve of bidegree (8,4), of genus 21, meets the elliptic curve exactly at the six points, with contact of order exactly four; smoothness by a dimension count, without Bertini's theorem.
• Adjoint jets — Lemma 3.2. Evaluation of adjoint sections on the subschemes of length at most four at the marked points has the same rank on the curve of genus 21 as on the elliptic curve; table of ranks 6, 12, 12, 10.
• The curve data — Proposition 3.3. Genus 21, deg L = 10, 4L = K, section dimensions h^0(L) = 2, h^0(2L) = 3, h^0(Z) = 1, h^0(2L+H) = 6, and surjective evaluation of (k+1)L on the marked points for k >= 2.
• The residue theorem — Theorem 2.2. The residues of a meromorphic differential on a smooth projective curve sum to zero; proved through the trace to P^1 and partial fractions.
• Realizable principal parts — Proposition 3.2. Principal parts along an effective divisor are realized by a global section exactly when they annihilate the jets of all sections of K - B under the residue pairing.
• Grothendieck's splitting theorem — Theorem 4.2. Every vector bundle on P^1 is a direct sum of line bundles O(a_i), with the integers a_i determined by the bundle.
• A perfect trace pairing — Proposition 5.1. For a finite map f: C -> P^1 and 2Lambda = K + 2H, the pairing Tr(T1 T2 / eta) on f_*Lambda is perfect and symmetric.
• The trace form — Lemma 1.1. f_*Lambda splits as O(3) + O^2 + O(-3); the trace form induces a nondegenerate form on a two-dimensional quotient W, with explicit squares of the evaluations at the marked points.
• Comparison with the elliptic curve — Lemma 2.1. Principal parts along twice the marked divisor identify W with the sections of Phi on the elliptic curve; the dual map is an isometry, sign and scalar included.
• The residue data — Proposition 3.2. A pair of meromorphic sections of 2L with simple poles at the six points whose local data satisfy p(0) ^ R(0) = 1 and sum (W_j - 1) = -9 deg L.
• Splitting along the marked points — Lemma 2.2. A cocycle of sections meromorphic at the marked points splits into an algebraic outside part and holomorphic inside parts exactly when its residues against all sections of K - B vanish.
• The second-order obstruction vanishes — Lemma 4.1. The four residue tests of the order-two vertical mismatch vanish, the totals being -(3/4) epsilon_ij deg L inside and outside.
• Finite compatible jets — Proposition 3.1. Finitely many gluing changes of orders 1 to 7 and two vector fields matching through index seven, with wedge t^5 (1 + O(t)) d/dt ^ d/dz.
• Realization — Lemma 1.1. The gluing changes define a Hausdorff complex surface containing the curve with self-intersection -10, and the jets give sections modulo the ninth power of the ideal.
• Graded vanishing — Lemma 2.1. H^1 of the graded pieces of I^9 F vanishes, so H^1 of every finite quotient of I^9 F vanishes.
• An analytic counterexample (assuming N1-N5) — Proposition 4.1. A normal surface germ, singular at one point, with free derivation sheaf of rank two; resolution by a curve of genus 21 and self-intersection -10. Uses five named theorems of Grauert, Cartan and the extension theorem on normal surfaces.
• Finite determinacy with a relation matrix — Lemma 1.1. If T^1 has finite length, equations g close to f together with an exact relation matrix close to that of f define the same complete local ring, up to an automorphism and a change of generators.
• Algebraization of an isolated germ (assuming A1, A2) — Proposition 3.1. A normal analytic germ with isolated singularity and free derivations has an affine normal algebraic model with the same completion and free derivation module.
• The counterexample (assuming N1-N5, A1, A2) — Theorem 4.1. A finitely generated normal C-algebra A of dimension two with Der(A) free of rank two and a nonregular maximal ideal: the Lipman-Zariski conjecture fails in characteristic zero.

Reading order

Nine lessons, to be read in order. The first states the problem and proves the characteristic-p and higher-dimensional examples. Lessons 2 to 5 build the curve of genus 21 and its residue data; lesson 4 proves the residue theorem, realizable principal parts and Grothendieck's splitting theorem on P^1. Lessons 6 and 7 glue the vector fields through order seven. Lesson 8 realizes the surface and contracts the curve, and lesson 9 algebraizes the singularity. Seven classical theorems are used without proof in the programme and are stated where they are used: Grauert's contraction theorem, Grauert's direct image theorem, the analytic theorem on formal functions, Theorem B on Stein spaces, the extension theorem on normal surfaces, Artin's approximation theorem, and the excellence of convergent power series rings.