A wall crossing formula for degrees of real central projections

The main result is a wall crossing formula for central projections defined on submanifolds of a real projective space. Our formula gives the jump of the degree of such a projection when the center of the projection varies. The fact that the degree depends on the projection is a new phenomenon, specific to real algebraic geometry. We illustrate this phenomenon in many interesting situations. The crucial assumption on the class of maps we consider is relative orientability, a condition which allows us to define a $\Z$-valued degree map in a coherent way. We end the article with several examples, e.g. the pole placement map associated with a quotient, the Wronski map, and a new version of the real subspace problem.Comment: 29 pages. First revised version: The proof of the "wall-crossing formula" is now more conceptional. We prove new general properties of the set of values of the degree map on the set of central projections. Second revised version: minor corrections. To appear in International Journal of Mathematic

Seiberg-Witten invariants for manifolds with b+=1b_+=1, and the universal wall crossing formula

In this paper we describe the Seiberg-Witten invariants, which have been introduced by Witten, for manifolds with $b_+=1$. In this case the invariants depend on a chamber structure, and there exists a universal wall crossing formula. For every K\"ahler surface with $p_g=0$ and $q$=0, these invariants are non-trivial for all $Spin^c(4)$-structures of non-negative index.Comment: LaTeX, 27 pages. To appear in Int. J. Mat

Real determinant line bundles

This article is an expanded version of the talk given by Ch. O. at the Second Latin Congress on "Symmetries in Geometry and Physics" in Curitiba, Brazil in December 2010. In this version we explain the topological and gauge-theoretical aspects of our paper "Abelian Yang-Mills theory on Real tori and Theta divisors of Klein surfaces".Comment: LaTeX, 8 page

Master Spaces for stable pairs

We construct master spaces for oriented torsion free sheaves coupled with morphisms into a fixed reference sheaf. These spaces are projective varieties endowed with a natural \C^*-action. The fixed point set of this action contains the moduli space of semistable oriented torsion free sheaves and the quot scheme associated with the given data. In the case of curves with trivial reference sheaf, our master spaces compactify the moduli spaces constructed by Bertram, Daskalopoulos and Wentworth. In the 2-dimensional case with trivial rank 1 reference sheaf, master spaces provide algebraic analoga of compactified moduli spaces of twisted quaternionic monopoles.Comment: 26 pages. New introduction and applications LaTeX2

Intrinsic signs and lower bounds in real algebraic geometry

A classical result due to Segre states that on a real cubic surface in ℙ ℝ 3 $\mathbb {P}^3_\mathbb {R}$ there exist two kinds of real lines: elliptic and hyperbolic lines. These two kinds of real lines are defined in an intrinsic way, i.e., their definition does not depend on any choices of orientation data. Segre's classification of smooth real cubic surfaces also shows that any such surface contains at least 3 real lines. Starting from these remarks and inspired by the classical problem mentioned above, our article has the following goals: (a) We explain a general principle which leads to lower bounds in real algebraic geometry. (b) We explain the reason for the appearance of intrinsic signs in the classical problem treated by Segre, showing that the same phenomenon occurs in a large class of enumerative problems in real algebraic geometry. (c) We illustrate these principles in the enumerative problem for real lines in real hypersurfaces of degree 2m-3$2m-3$ in ℙ ℝ m $\mathbb {P}^m_\mathbb {R}