Maurizio Cornalba

Geometry of algebraic curves

Preface.- Guide to the Reader.- Chapter IX. The Hilbert Scheme.- Chapter X. Nodal curves.- Chapter XI. Elementary deformation theory and some applications.- Chapter XII. The moduli space of stable curves.- Chapter XIII. Line bundles on moduli.- Chapter XIV. The projectivity of the moduli space of stable curves.- Chapter XV. The Teichmuller point of view.- Chapter XVI. Smooth Galois covers of moduli spaces.- Chapter XVII. Cycles on the moduli spaces of stable curves.- Chapter XVIII. Cellular decomposition of moduli spaces.- Chapter XIX. First consequences of the cellular decomposition .- Chapter XX. Intersection theory of tautological classes.- Chapter XXI. Brill-Noether theory on a moving curve.- Bibliography.- Index.

Calculating cohomology groups of moduli spaces of curves via algebraic geometry

We compute the first, second, third, and fifth rational cohomology groups of the moduli space of stable n-pointed genus g curves, for all g and n, using (mostly) algebro-geometric techniques.

Analytic Cycles and Vector Bundles on Non-Compact Algebraic Varieties.

A. Orders of Growth on Algebraic Varieties . . . . . . . . . . . 4 w 1. Review of the Classical Theory . . . . . . . . . . . . . 4 w 2. Generalization to Algebraic Varieties . . . . . . . . . . . 6 w 3. Exhaustion Functions and K~ihler Metrics on Special Affine Varieties . . . . . . . . . . . . . . . . . . . . . . . 9 84. Order of Growth of Analytic Sets . . . . . . . . . . . . 13 w 5. Order of Growth of Holomorphic Mappings; the First Main Theorem of Nevanlinna Theory . . . . . . . . . . . . . 15 w 6. Orders of Growth of Holomorphic Vector Bundles . . . . . 17

Moduli of roots of line bundles on curves

We treat the problem of completing the moduli space for roots of line bundles on curves. Special attention is devoted to higher spin curves within the universal Picard scheme. Two new different constructions, both using line bundles on nodal curves as boundary points, are carried out and compared with pre-existing ones.

Algebraic geometry and path integrals for closed strings

Abstract The p th loop contribution to the partition function for closed strings is studied by applying recent mathematical results on the geometry of the moduli space M p of smooth algebraic curves of genus p . By reasoning on determinants of operators and line bundles over M p , we get a geometric explanation of the critical dimensions 26 and 10. The extension of path integrals for strings to the compactified moduli space M p of stable curves is also discussed. While the Weil-Peterson measure has a continuous extension on M p , the bosonic path integral has a bad behaviour on the boundary M p − M p . Instead, the functional approach to the spinning string of Ramond-Neveu-Schwarz seems to yield a finite p th loop contribution to the partition function.

How to get around a simple quadratic fold

SummaryWe analyse from a theoretical point of view a computational technique previously introduced [2, 5] for tracing branches of solutions of nonlinear equations near simple quadratic folds.

The Picard group of the moduli stack of stable hyperelliptic curves

We compute the Picard group of the moduli stack of stable hyperelliptic curves of any genus, exhibiting explicit and geometrically meaningful generators and relations.

Teichmüller space via Kuranishi families

In this partly expository note we construct Teichm¨ uller space by patching together Kuranishi families. We also discuss the basic properties of Te- ichmspace, and in particular show that our construction leads to simplifica- tions in the proof of Teichmtheorem asserting that the genus g Teichm¨ uller

The cephalic labial gland secretions of two socially parasitic bumblebees Bombus hyperboreus (alpinobombus) and Bombus inexspectatus (thoracobombus) question their inquiline strategy

Social parasitic Hymenopterans have evolved morphological, chemical, and behavioral adaptations to overcome the sophisticated recognition and defense systems of their social host to invade host nests and exploit their worker force. In bumblebees, social parasitism appeared in at least 3 subgenera independently: in the subgenus Psithyrus consisting entirely of parasitic species, in the subgenus Alpinobombus with Bombus hyperboreus, and in the subgenus Thoracobombus with B. inexspectatus. Cuckoo bumblebee males utilize species‐specific cephalic labial gland secretions for mating purposes that can impact their inquiline strategy. We performed cephalic labial gland secretions in B. hyperboreus, B. inexspectatus and their hosts. Males of both parasitic species exhibited high species specific levels of cephalic gland secretions, including different main compounds. Our results showed no chemical mimicry in the cephalic gland secretions between inquilines and their host and we did not identify the repellent compounds already known in other cuckoo bumblebees.

Enumerative Geometry of Curves

In this chapter we will continue in the direction suggested by Sections 4 and 5 of the previous chapter: that is, we will try to solve some of the enumerative problems that arise in the theory of curves and linear systems. While this is in some sense a quantitative approach, qualitative results may also emerge. For example, the answer to the enumerative question: “How many g d r ’s does a curve C possess” (Theorem (4.4) in Chapter VII) implies the existence theorem (Theorem (2.3) in Chapter VII).