We investigate the spaces of rational curves on a general hypersurface. In particular, we show that for a general degree d hypersurface in P n with n ≥ d + 2, the space M 0,0 (X, e) of degree e Kontsevich stable maps from a rational curve to X is an irreducible local complete intersection stack of dimension e(n − d + 1) + n − 4. This resolves all but one case of a conjecture of Coskun, Harris and Starr, and also proves that the Gromov-Witten invariants of these hypersurfaces are enumerative.

Let [Formula: see text] be a general Fano complete intersection of type [Formula: see text]. If at least one [Formula: see text] is greater than [Formula: see text], we show that [Formula: see text] contains rational curves of degree [Formula: see text] with balanced normal bundle. If all [Formula: see text] are [Formula: see text] and [Formula: see text], we show that [Formula: see text] contains rational curves of degree [Formula: see text] with balanced normal bundle. As an application, we prove a stronger version of the theorem of Tian [27], Chen and Zhu [4] that [Formula: see text] is separably rationally connected by exhibiting very free rational curves in [Formula: see text] of optimal degrees.

Abstract. Let b• be a sequence of integers 1 < b1 ≤ b2 ≤ · · · ≤ bn−1. Let Me(b•) be the space parameterizing nondegenerate, immersed, rational curves of degree e in P n such that the normal bundle has the splitting type n−1 i=1 O(e + bi). When n = 3, celebrated results of Eisenbud, Van de Ven, Ghione and Sacchiero show that Me(b•) is irreducible of the expected dimension. We show that when n ≥ 5, these loci are generally reducible with components of higher than the expected dimension. We give examples where the number of components grows linearly with n. These generalize an example of Alzati and Re. IntroductionRational curves play a central role in the birational and arithmetic geometry of projective varieties. Consequently, understanding the geometry of the space of rational curves is of fundamental importance. The local structure of this space is governed by the normal bundle. In this paper, we study the dimensions and irreducible components of the loci in the space of rational curves in P n parameterizing curves whose normal bundles have a specified splitting type. We work over an algebraically closed field of characteristic zero.We first set some notation. Let f : P 1 → P n be a nondegenerate, unramified, birational map of degree e. Then the normal bundle N f defined byis a vector bundle of rank n−1 and degree e(n+1)−2. By Grothendieck's theorem, N f is isomorphic to a direct sum of line bundles. Let Mor e (P 1 , P n ) denote the morphism scheme parameterizing degree e morphisms f : P 1 → P n . Let b • denote an increasing sequence of integersdenote the locally closed locus in Mor e (P 1 , P n ) parameterizing nondegenerate, unramified morphisms of degree e such thatThe scheme Mor e (P 1 , P n ) is irreducible of dimension (n + 1)(e + 1) − 1. The codimension of the locus of vector bundles E on P 1 with a specified splitting type in the versal deformation space is given by h 1 (P 1 , End(E)) [C08, Lemma 2.4]. In analogy, we say that the expected codimension of M e (b • ) is h 1 (P 1 , End(N f )). Equivalently, the expected dimension is (e + 1)(n + 1) − 1 − h 1 (P 1 , End(N f )). [R93] show that the locus of nondegenerate rational curves with a specified splitting type for f * T P n is irreducible of codimension h 1 (P 1 , End(f * T P n )) for all n ≥ 3. The behavior of M e (b • ) for n ≥ 5 is in stark contrast to these results.Recently, Alzati and Re [AR17] showed that the locus of rational curves of degree 11 in P 8 whose normal bundles have the splitting type O(13) 3 ⊕ O(14) 2 ⊕ O(15) 2 is reducible. This was the first indication that the geometry of M e (b • ) is much more complicated for large n. This paper grew out of our attempt to generalize their example. We produce examples of reducible M e (b • ) in P 5 with e < 11, we find M e (b • ) with arbitrarily many components, and show that the difference between the expected dimension and actual dimension of a component of M e (b • ) can grow arbitrarily large.We now summarize our results in greater detail. First, following Sacchiero [Sa80], we explain ...

Originally introduced by Chalupa, Leath and Reich for use in modeling disordered magnetic systems, $r$-bootstrap percolation is the following deterministic process on a graph.  Given an initial infected set, vertices with at least $r$ infected neighbors are infected until no new vertices can be infected.  A set percolates if it infects all the vertices of the graph, and a percolating set is minimal if no proper subset percolates.  We consider minimal percolating sets in finite trees.  We show that if $A$ is a minimal percolating set on a tree $T$ with $n$ vertices and $\ell$ vertices of degree less than $r$ (leaves in the case $r=2$), then $\frac{(r-1)n+1}{r} \leq |A| \leq \frac{rn+\ell}{r+1}$.  Moreover, we show that the difference between the sizes of a largest and smallest minimal percolating sets is at most $\frac{(r-1)(n-1)}{r^2}$.  Finally, we describe $O(n)$ algorithms for computing the largest (for $r=2$) and smallest (for $r \geq 2$) minimal percolating sets.

Let X be a smooth projective surface of irregularity 0. The Hilbert scheme of n points on X parameterizes zero-dimensional subschemes of X of length n. In this paper, we discuss general methods for studying the cone of ample divisors on the Hilbert scheme. We then use these techniques to compute the cone of ample divisors on the Hilbert scheme for several surfaces where the cone was previously unknown. Our examples include families of surfaces of general type and del Pezzo surfaces of degree 1. The methods rely on Bridgeland stability and the Positivity Lemma of Bayer and Macri.Comment: 18 pages, comments welcome