Spaces of rational curves on complete intersections (Q2845018)

This paper studies the space of rational curves on a general complete intersection.NEWLINENEWLINEThe main results are:NEWLINENEWLINETheorem 1.3. Let \(X\) be a general hypersurface of degree \(d \leq n-1\) in \(\mathbb{P}^n\). If \(\frac{(n-d)(n-d-1)}{2}>n-1\), then the evaluation map \(\mathrm{ev}: \overline{\mathcal{M}}_{0,1}(X, 2) \to X\) is flat of relative dimension \(2n-2d\).NEWLINENEWLINETheorem 1.4. Let \(X\) be a general hypersurface of degree \(d <(2n+2)/3, n \geq 23\) in \(\mathbb{P}^n\). Then for every \(e \geq 1\), the evaluation map \(\mathrm{ev}: \overline{\mathcal{M}}_{0,1}(X, e) \to X\) is flat of relative dimension \(e(n+1-d)-2\). And \(\overline{\mathcal{M}}_{0,0}(X, e)\) is an integral local complete intersection stack of expected dimension \(e(n+1-d)+n-4\).NEWLINENEWLINEThe key ingredient in the proof is a study of proper family of embedded conics.NEWLINENEWLINEThe authors also prove some results about the non-free lines on a general complete intersection.