Rational curves on hypersurfaces (Q5891548)

Let \(R_e(X)\) be the family of smooth rational curves of degree \(e\) on a hypersurface \(X\) of degree \(d\). The expected dimension, denoted by \(\mu\), of \(R_e(X)\) is equal to the Euler characteristic of the normal bundle on a smooth rational curve. In the paper under review, the author studies \(R_e(X)\) without restriction on the characteristic of the ground field. Under some conditions on \(d,e\) and \(\mu\), it is shown that \(R_e(X)\) is empty, smooth, or connected for general \(X\).