Algebraic Geometry

We construct a canonical free resolution for arbitrary monomial modules and lattice ideals. This includes monomial ideals and defining ideals of toric varieties, and it generalizes our joint results with Irena Peeva for generic ideals.

This is a first graduate course in algebraic geometry. It aims to give the student a lift up into the subject at the research level, with lots of interesting topics taken from the classification of surfaces, and a human-oriented discussion of some of the technical foundations, but with no pretence at an exhaustive treatment. The early chapters introduce topics that are useful throughout projective and algebraic geometry, make little demands, and lead to fun calculations. The intermediate chapters introduce elements of the technical language gradually, whereas the later chapters get into the substance of the classification of surfaces. Special features include the theory of minimal models of surfaces via Mori theory, a complete selfcontained proof of the theorems on classification of surfaces, and a clean treatment of the foundational results on rational and elliptic Gorenstein surface singularities. Contents: Chapter 1. The cubic surface p.4 Exercises to Chapter 1 p.12 Chapter 2. Rational scrolls p.14 Exercises to Chapter 2 p.23 Chapter A. Curves on surfaces and intersection numbers p.26 Exs to Ch A p.35 Chapter B. Sheaves and coherent cohomology p.37 Exercises to Chapter B p.47 Chapter C. Guide to the classification of surfaces 51 Chapter 3. K3s p.63 Exercises to Chapter 3 p.76 Chapter 4. Surfaces and singularities p.80 Exercises to Chapter 4 p.106 Chapter D. Minimal models of surfaces via Mori theory p.110 Chapter E. Proof of the classification of surfaces p.121 References p.146

Consider the blow up π: X ˜ →X of a rational surface X at a point. Let V ˜ be a holomorphic bundle over X ˜ whose restriction to the exceptional divisor equals ${\cal{O}(j) \oplus {\cal O}(-j)$ and define V=( π ∗ V ˜ ) ∨∨ . Friedman and Morgan gave the following bounds for the second Chern classes j≤ c 2 ( V ˜ )− c 2 (V)≤ j 2 . We show that these bounds are sharp.

In this paper we introduce a formula to compute Chern classes of fibered products of algebraic surfaces. For f, a generic projection to CP^2, of an algebraic surface X, we define X_k (for k smaller than degf) to be the k products of X over f minus the big diagonal. For k=degf, X_k is called the Galois cover of f w.r.t. full symmetric group. Let S be the branch curve of f. We give a formula for c_1^2 and c_2 of X_k, in terms of degf, degS, and the number of cusps, nodes and branch points of S. We apply the formula in 2 examples and add a conjecture concerning the spin structure of fibered products of Veronese surfaces.

We prove a simple formula for MacPherson's Chern class of hypersurfaces in nonsingular varieties. The result highlights the relation between MacPherson's class and other definitions of homology Chern classes of singular varieties, such as Mather's Chern class and a class introduced by W. Fulton.

We classify the log del Pezzo surface (S,B) of rank 1 with no 1-,2-,3-,4-, or 6-complements with the additional condition that B has one irreducible component C which is an elliptic curve, and that C has the coefficient b in B with (1/n)floor((n+1)b)=1 for n=1,2,3,4, and 6.

Let X be a codimension two nonsingular subvariety of a nonsingular quadric $\Q{n}$ of dimension n≥5 . We classify such subvarieties when they are scrolls. We also classify them when the degree d≤10 . Both results were known when n=4 . Keywords: Classification; liaison; low codimension; low degree; quadric; scroll; vector bundle

This paper explicitly describes Hodge structures of complete intersections of ample hypersurfaces in compact simplicial toric varieties.

A classification of commutative integral domains consisting of ordinary differential operators with matrix coefficients is established in terms of morphisms between algebraic curves.

We investigate the existence and non-existence of modular forms of low weight with a character with respect to the paramodular group Γ t and discuss the resulting geometric consequences. Using an advanced version of Maaß\ lifting one can construct many examples of such modular forms and in particular examples of weight 3 cusp forms. Consequently we find many abelian coverings of low degree of the moduli space ${\Cal A}_t$ of (1,t)-polarized abelian surfaces which are not unirational. We also determine the commutator subgroups of the paramodular group Γ t and its degree 2 extension Γ + t . This has applications for the Picard group of the moduli stack ${\underline{\Cal A}}_t$. Finally we prove non-existence theorems for low weight modular forms. As one of our main results we obtain the theorem that the maximal abelian cover ${\Cal A}_t^{com}$ of ${\Cal A}_t$ has geometric genus 0 if and only if t=1, 2, 4 or 5. We also prove that ${\Cal A}_t^{com}$ has geometric genus 1 for t=3 and 7.