Spinor states of real rational curves in real algebraic convex 3-manifolds and enumerative invariants

By Jean-Yves Welschinger. 2003

102 citations recorded by the source corpus.

Let X be a real algebraic convex 3-manifold whose real part is equipped with a Pin i structure. We show that every irreducible real rational curve with non-empty real part has a canonical spinor state belonging to f§1g. The main result is then that the algebraic count of the number of real irreducible rational curves in a given numerical equivalence class passing through the appropriate number of points does not depend on the choice of the real conflguration of points, provided that these curves are counted with respect to their spinor states. These invariants provide lower bounds for the total number of such real rational curves independantly of the choice of the real conflguration of points.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/cb0a7a54f2414739f28b52323ce01f1296e17094

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In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions (2006, methodology): “In [21], Welschinger does exactly that using the splitting of N x into holomorphic line bundles.”
Welschinger invariant and enumeration of real rational curves (2003, background): “3. When we were finishing these notes, we have become acquainted with a new preprint of Welschinger [ 14 ], where he proves the invariance of another characteristic, specific for rational curves in real algebraic convex 3-folds.”
Enumeration of curves via floor diagrams (2007, unknown): “Nevertheless, Welschinger proved in [7], [8] that in the case when n = 2, 3, g = 0 and lj = 0 for any j > 0, there is a consistent choice of signs ±1 such that the number of the corresponding real curves counted with the sign depends only on d (note that the assumption lj = 0 for j > 0 allows us to ”