(a) Given any finitely generated group \( G \subset {\rm {Cl}} R \), is there a geometric local ring \(B\) with \( \hat B \cong R \) and \( G = {\rm {Cl}} B \)?

(b) Given \( \alpha_1, \dots, \alpha_r \in {\rm {Cl}} R \), is there a geometric local ring \( B \) with \( \hat B \cong R \) and \( \alpha_i \in {\rm {Cl}} B \) for each \( i \)?

(c) Is every \( \alpha \in {\rm {Cl}} R \) a geometric divisor?

(d) Do geometric divisors form a subgroup of \( {\rm {Cl}} R \)?

We have easy implications \( (a) \Rightarrow (b) \Rightarrow (c) \Rightarrow (d) \) and conjecture that if \( R \) is the completion of a normal geometric local complete intersection ring and \( G \subset {\rm {Cl}} R \) is any finitely generated subgroup, then there is a geometric local ring \( B \) with \( \hat B \cong R \) and \( G = {\rm {Cl}} B \subset {\rm {Cl}} R \). We make some progress toward this conjecture with the following result:
Theorem 1. Let \( p \in V \subset \mathbb P^n \) be a normal complete intersection point on an complex variety (take \( n=3 \) if \( \dim V = 2 \)). Then for any finitely generated subgroup \( G \subset {\rm {Cl}} {\mathcal O}_{V,p} \) there is a geometric local ring \( B \) with \( \hat B \cong \hat {\mathcal O}_{V,p} \) and \( {\rm {Cl}} B = G \subset {\rm {Cl}} {\mathcal O}_{V,p} \subset {\rm {Cl}} \hat {\mathcal O}_{V,p} \).
As an application we show that every finitely generated group appears as the class group of a local ring of a normal point on some complex algebraic variety.

Theorem. Let \( X \subset \mathbb P^N \) be a normal projective variety of dimension \( \geq 4 \) and let \( Z \subset X \) be a closed subscheme of codimension \( \geq 2 \) with \( {\mathcal I}_Z (d-1) \) generated by global sections. Assume that the irreducible components \( Z_i \subset Z \) of codimension two are not contained in the the singular locus of \( X \) and have generic embedding dimension at most \( \dim X - 1 \). Then the general member \( Y \in |H^0({\mathcal I}_Z (d))| \) is normal and the restriction map \( {\rm {Cl}} X \to {\rm {Cl}} Y \) is split injective with cokernel generated by the supports of the \( Z_i \).

As an application, we show that the completion of any normal complete intersection singularity of dimension at least three is the completion of a unique factorization domain.

1. What subgroups appear as the image of the map on divisor class groups \( {\rm {Cl}} A \hookrightarrow {\rm {Cl}} B \) taken over all geometric domains \( A \) whose completion is isomorphic to \( B \)?

2. Is it possible to obtain \( \langle \omega_B \rangle \subset {\rm {Cl}} B \) as such an image?

While there has been quite a bit of progress on problem 2, especially for complete intersection rings, almost no progress has been made on problem 1. In this paper we prove that for completions of local rings of surface rational double point singularities, the answer is all subgroups.

1. The Classical Theorem. Noether's original idea and statement from 1882 and the proof of Lefschetz from the 1920s, Moishezon's variant for algebraic homology classes (1967) and the generalization to finite characteristic due to Deligne, Grothendieck and Katz (SGA 7), and the stronger Grothendieck-Lefschetz theorem for higher dimensions.

2. New ideas: the infinitesimal approach. After reviewing variations of Hodge structures and some results about the moduli of projective hypersurfaces, we present the infinitesimal Noether theorem of Carlson, Green, Griffiths and Harris, the explicit Noether theorem (via Koszul cohomology) of Green, the degeneration proof of Griffiths and Harris, Ein's generalization of Noether's theorem to dependency loci of sections of vector bundles, Joshi's variant for singular surfaces, recent results of Ravindra and Srinivas about the class groups with normal ambient spaces, and our own variation of Noether's theorem with base locus.

3. Components of the Noether-Lefschetz locus. The density of general components of Ciliberto, Harris, Miranda and Green, the work of Green and Voisin on components of small codimension and Otwinowska's asymptotic result, the work of Ciliberto and Lopez on the distribution of codimensions, and Voisin's counterexample to the conjecture that there are finitely many special components.

4. An application. A recent application of Noether-Lefschetz theory to a question of V. Srinivas about class groups of local rings.

Theorem. Let \( Z \subset \mathbb P^3 \) be a closed subscheme which lies properly on a normal surface. Then the general surface \( S \) of high degree containing \( Z \) is normal and its class group is freely generated by \( {\mathcal O}_S (1) \) and the supports of the curve components of \( Z \).

As special cases, we recover the classical Noether-Lefschetz theorem by taking \( Z \) to be empty, concluding that the general surface \( S \subset \mathbb P^3 \) of degree \( d > 3 \) is smooth with Picard group isomorphic to \( \mathbb Z \), generated by the hyperplane class. When \( Z \) is a smooth connected curve we recover a theorem of Lopez which has found many applications [Lopez91]. When \( Z \) is zero dimensional and non-reduced, we improve on Joshi's result that the general singular surface \( S \) satisfies the conclusion of the classic theorem [Joshi95]. There are also several applications to computing Picard groups.

An \( (n-1) \)-dimensional submanifold \( \subset \mathbb R^n \) nicely bounds a closed subset \( D \subset \mathbb R^n \) if \( D \) is the boundary of \( A \), each connected component of the closure of \( A \) contains exactly one connected component of \( D \), and \( D \) is be the closure of an \( (n-2) \)-dimensional submanifold \( D_0 \subset \mathbb R^n \) with singular locus \( S = D - D_0 \) of codimension at least 4 with the closure of \( A \) locally diffeomorphic to a half-space along \( D_0 \). The first result is the following:

Theorem 1. Let \( D \subset \mathbb C^n \) be a smooth connected non-bifurcated hypersurface and let \( X \) be a simply connected manifold. Suppose that \( F: X \to \mathbb C^n \) is a local diffeomorphism, and the restriction \( F: X - F^{-1} (D) \to \mathbb C^n - D \) is a d-sheet covering map. then \( d = 1 \) or \( d = \infty \).

As a corollary we show that if \( F: \mathbb C^n \to \mathbb C^n \) is a polynomial map with nonvanishing Jacobian and the number of points in all fibers over the points outside \( D \) which is a non-bifurcated hypersurface in \( \mathbb C^n \) are equal to the degree of \( F \), then F is an automorphism. Over the reals we prove the following:

Theorem 2. let \( F : X \to \mathbb R^n \) be a local diffeomorphism, \( X \) a connected manifold with \( H_1(X,Z)=0 \), and \( D \subset \mathbb R^n \) a closed set for which the restriction \( F: X - F^{-1} (D) \to \mathbb R^n - D \) is a d-sheeted covering. If \( D \) is nicely bounded by a connected \( (n-1) \)-dimensional submanifold \( A \) such that \( \mathbb R^n - A \) is simply connected, then \( d = 1,2 \) or \( \infty \).

In fact, each of these values of \( d \) can be achieved (the interesting case \( d=2 \) is achieved by an collared embedding of the Klein bottle in \( \mathbb R^4 \). The proof uses algebraic topology and surgery theory.

Theorem. If \( X \) is a simply connected manifold and

(a) the closure of \( D \) in \( \mathbb P^n \) meets the hyperplane at infinity transversely

(b) \( D \) has at worst normal crossing singularities away from a set of codimension 3

The result is sharp in that both cases actually occur: \( d=1 \) is achieved by the identity map while \( d=\infty \) is achieved by the complex exponential map. For algebraic maps, \( d=1 \) is the only possibility, giving an application to the Jacobian conjecture, though note that if the conjecture is true, then \( D \) may be taken empty. The proof uses methods of algebraic topology and Fulton's solution to the Zariski conjecture [Fulton80].

1. Geography of the Hilbert scheme. Here I explained work of Hartshorne, Martin-Deschamps and Perrin which describe the pairs \( (d,g) \) for which \( H(d,g) \) is non-empty.

2. Connectedness Results. This section describes known results of several authors regarding connectedness of flat families of space curves, including Hartshorne's thesis [Hartshorne66] and Lawrence Ein's result about nonspecial curves [Ein86].

3. The Hilbert scheme \( H(4,-99) \). In an effort to convey the state of the art of this difficult problem, I described the contents of my paper with Enrico Schlesinger [Hilbert schemes of degree four curves, Compositio Math. 139 (2003)] and as an example described the Hilbert scheme \( H(4,-99) \) and its \( 529 \) irreducible components.

4. Curves on the double quadric. This section describes in detail a new technique for constructing certain families of curves from section 3, based on another paper with Enrico Schlesinger [Curves on a double surface, Collectanea Math. 54 (2003)].

5. Subcanonical curves. Here I classified the subcanonical curves appearing in \( H(4,-99) \). There are \( 18 \) such families, one of which contains subcanonical curves that were discovered by Valabrega and Chiantini 20 years earlier [CV84].

In the case of curves in \(\mathbb P^3\), this theorem recovers as corollaries the results in papers of Sauer [Sauer85], Gruson and Peskine [GP78] and Maggioni and Ragusa [MR88]. In fact, our examples of sharpness for the theorem improve on these papers in that (a) the method works in arbitrary characteristic, (b) the curves lie on smooth surfaces of minimal degree and (c) the curves have maximal order multisecant lines. Our construction uses some local algebra which concretely describes the singular locus of a blow-up along a codimension two subvariety, recovering a result of O'Carroll and Valla [OV97]. Another consequence is that general arithmetically Cohen-Macaulay subscheme of codimension two has singular locus of codimension \(> 3\), as shown earlier by Chang's filtered Bertini theorem [Chang89].

\( \dagger \) It was from Fernando (Ferran) Serrano that I first learned algebraic geometry at U.C. Berkeley while he was a postdoc there in 1988.

In this work we prove a bound on the higher cohomology of the tangent and cotangent bundles of a smooth minimal threefold with ample canonical bundle to give a quadratic bound on the third Chern number in terms of the first two Chern numbers. Chang and Kim improve these bounds under the additional assumption that threefolds have principal Picard group generated by a very ample divisor [CK00].