Normal Bundles of Rational Curves in Projective Spaces

Minimal Rational Curves in Moduli Spaces of Stable Bundles

Let C be a smooth projective curve of genus g ≥ 2 and L be a line bundle on C of degree d. Assume that r ≥ 2 is an integer coprime with d. Let M := UC(r,L) be the moduli space of stable vector bundles on C of rank r and with the fixed determinant L. It is well-known that M is a smooth projective Fano variety with Picard number 1. For any projective curve in M , we can define its degree with res...

Counting Rational Curves of Arbitrary Shape in Projective Spaces

Counting rational curves of arbitrary shape in projective spaces

We present an approach to a large class of enumerative problems concerning rational curves in projective spaces. This approach uses analysis to obtain topological information about moduli spaces of stable maps. We demonstrate it by enumerating one-component rational curves with a triple point or a tacnodal point in the three-dimensional projective space and with a cusp in any projective space. ...

Enumeration of One-Nodal Rational Curves in Projective Spaces

We give a formula computing the number of one-nodal rational curves that pass through an appropriate collection of constraints in a complex projective space. The formula involves intersections of tautological classes on moduli spaces of stable rational maps. We combine the methods and results from three different papers.

On the Normal Bundles of Smooth Rational Space Curves

in this note we consider smooth rational curves C of degree n in threedimensional projective space IP 3 (over a closed field of characteristic 0). To avoid trivial exceptions we shall always assume that n ~ 4 (this does not hold however for certain auxiliary curves we shall consider). Let N = N c be the normal bundle of C in IP 3. Since degel(IP3)=4, and d e g c l ( l P 0 = 2 , we have that d e...