Results on completed cones, elementary modifications, Chern class inequalities and a Kummer surface

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.

• The completed cone over a surface — Proposition 3.1. For a smooth integral projective surface X over C with H ample and globally generated, the section ring T(X,H) is a domain finite over C[x_0,x_1,x_2], and its completion R(X,H) at the vertex is a complete Noetherian local domain of dimension three with residue field C, finite over C[[x_0,x_1,x_2]].
• The regular model and normality — Proposition 4.1. The total space of O_X(-H), base-changed to C[[x_0,x_1,x_2]], is a regular integral scheme W, proper over Spec R(X,H) and isomorphic to the punctured spectrum off the zero section, with Gamma(W, O_W) = R(X,H); hence R(X,H) is normal.
• Small Cohen–Macaulay modules are free over the power series ring — Lemma 5.1. A small Cohen-Macaulay module over R(X,H) is free over C[[x_0,x_1,x_2]], has positive rank, and is locally free on the punctured spectrum.
• Extension to the regular model — Proposition 6.1. A small Cohen-Macaulay module of rank r gives a torsion-free sheaf E on W whose restriction P to the exceptional surface is torsion-free of rank r; the pushforward of E is trivial of rank N = rh off the exceptional plane and restricts to f_*P on it.
• Chern classes on the plane through Hilbert polynomials — Lemma 1.1. Rank, c_1 and ch_2 of coherent sheaves on P^2, defined as coordinates of the Hilbert polynomial, are additive and transform correctly under twists; c_1 >= 0 for sheaves of rank zero and for globally generated sheaves.
• Harder–Narasimhan filtrations — Proposition 2.2. Assume (CS). Every torsion-free sheaf of positive rank on a smooth projective surface with an ample H has a filtration with torsion-free semistable factors of strictly decreasing slopes.
• Balancing by elementary modifications — Proposition 3.2. Assume (CS). A torsion-free sheaf on an integral Noetherian scheme, with torsion-free restriction to a Cartier divisor that is a smooth projective surface with ample conormal bundle, can be modified along the divisor so that the Harder-Narasimhan slopes of the restriction lie in an interval of length at most one.
• A variance inequality — Proposition 4.2. If L' is torsion-free on an integral Noetherian scheme, trivial of rank N off a Cartier divisor D = P^2 whose ideal restricts to O(1), then F = L'|_D satisfies 2N ch_2(F) >= c_1(F)^2.
• One semistable factor — Lemma 2.1. Assume (CS), (NF) and (BI). For a finite map f: X -> P^2 of degree h with f^*O(1) = O_X(H) and a torsion-free semistable sheaf G on X, v(f_*G) <= 1/4 - (K_X^2 - 2c_2(X))/(12h).
• The numerical obstruction — Theorem 3.3. Assume (CS), (NF) and (BI). If the completed cone R(X,H) over a smooth projective complex surface, with H ample and globally generated, has a small Cohen-Macaulay module, then K_X^2 - 2c_2(X) <= 6H^2.
• The complete quadrangle on the blown-up plane — Lemma 2.1. Assume (BL). On the plane blown up at four points in general position, the six joining lines and the four exceptional curves form a simple normal crossing divisor with fifteen nodes, linearly equivalent to -2K_B, and K_B^2 = 5.
• A Kummer cover and its local form — Proposition 3.3. For a prime n, the normalization of the blown-up plane in the field obtained by adjoining n-th roots of five ratios of the linear forms is a smooth surface of degree n^5 over the base, locally of the form t -> t^n along the branch curves.
• Chern numbers of the cover — Proposition 4.3. Assume (BL), (EU) and (CS). With d = n^5: K_X^2 = 5d(1 - 2/n)^2, c_2(X) = d(2 - 10/n + 15/n^2), and K_X^2 - 2c_2(X) = d(1 - 10/n^2).
• An ample globally generated polarization — Proposition 5.3. Assume (BL), (EU) and (CS). The divisor H = 3J - E on the cover is ample and globally generated, with nH = pi^*(-K_B), H^2 = 5n^3 and K_X^2 - 2c_2(X) = n^3(n^2 - 10).
• A complete local domain without a small Cohen–Macaulay module — Theorem 6.1. Assume (CS), (NF), (BI), (BL) and (EU). For n = 7, R(X,H) is a three-dimensional complete Noetherian normal local domain containing C, with residue field C, without a nonzero finitely generated module of depth three; the same holds for the uncompleted local ring at the vertex (Corollary 6.2).

Reading order

Four lessons, to be read in order. The theorem is proved from five named hypotheses on smooth projective surfaces that are not proved in the programme: Chern classes and Riemann-Roch for coherent sheaves, Noether's formula, Bogomolov's inequality, the plane blown up at four points, and Euler numbers. Commutative algebra, schemes and cohomology are taken from other courses of the programme and are cited where they are used.