Spaces of rational curves in complete intersections

We prove that the space of smooth rational curves of degree e in a general complete intersection of multidegree (d1,...., dm) in Pn is irreducible of the expected dimension if Σi=1m di < 2n/3 and n is large enough. This generalizes the results of Harris, Roth and Starr, and is achieved by proving that the space of conics passing through any point of a general complete intersection has constant dimension if Σi=1m di is small compared to n

Fibers of Generic Projections

Let X be a smooth projective variety of dimension n in P^r. We study the fibers of a general linear projection pi: X --> P^{n+c}, with c > 0. When n is small it is classical that the degree of any fiber is bounded by n/c+1, but this fails for n >> 0. We describe a new invariant of the fiber that agrees with the degree in many cases and is always bounded by n/c+1. This implies, for example, that if we write a fiber as the disjoint union of schemes Y' and Y'' such that Y' is the union of the locally complete intersection components of Y, then deg Y'+deg Y''_red <= n/c+1 and this formula can be strengthened a little further. Our method also gives a sharp bound on the subvariety of P^r swept out by the l-secant lines of X for any positive integer l, and we discuss a corresponding bound for highly secant linear spaces of higher dimension. These results extend Ziv Ran's "Dimension+2 Secant Lemma".Comment: Proof of the main theorem simplified and new examples adde

Rational curves on del Pezzo surfaces in positive characteristic

We study the space of rational curves on del Pezzo surfaces in positive characteristic. For most primes p we prove the irreducibility of the moduli space of rational curves of a given nef class, extending results of Testa in characteristic 0. We also investigate the principles of Geometric Manin's Conjecture for weak del Pezzo surfaces. In the course of this investigation, we give examples of weak del Pezzo surfaces defined over $\mathbb{F}_{2}(t)$ or $\mathbb{F}_{3}(t)$ such that the exceptional sets in Manin's Conjecture are Zariski dense.Comment: minor changes, 43 pages, to appear in Transactions of the American Mathematical Society Series B

Spaces of rational curves in complete intersections

We prove that the space of smooth rational curves of degree $e$ in a general complete intersection of multidegree $(d_1, ..., d_m)$ in \PP^n is irreducible of the expected dimension if $\sum_{i=1}^m d_i <\frac{2n}{3}$ and $n$ is large enough. This generalizes the results of Harris, Roth and Starr \cite{hrs}, and is achieved by proving that the space of conics passing through any point of a general complete intersection has constant dimension if $\sum_{i=1}^m d_i$ is small compared to $n$

On the asymptotic enumerativity property for Fano manifolds

We study the enumerativity of Gromov-Witten invariants where the domain curve is fixed in moduli and required to pass through the maximum possible number of points. We say a Fano manifold satisfies asymptotic enumerativity if such invariants are enumerative whenever the degree of the curve is sufficiently large. Lian and Pandharipande speculate that every Fano manifold satisfies asymptotic enumerativity. We give the first counterexamples, as well as some new examples where asymptotic enumerativity holds. The negative examples include special hypersurfaces of low Fano index and certain projective bundles, and the new positive examples include many Fano threefolds and all smooth hypersurfaces of degree $d \leq (n+3)/3$ in $\mathbb{P}^n$.Comment: comments welcome

An Investigation of Family Therapy-Based Training on an Adolescent\u27s Self-Harm and Character Strengths

Background: The prevalence of non-suicidal self-injury has increased in adolescents. This study aimed to assess the effectiveness of family therapy compared with treatment as usual in improving character strengths and reducing self-harm repetition in adolescents. Methods: The research was designed as semi-experimental with a pre-test, post-test, and control group. The statistical population was all students of Tehran\u27s middle schools with self-injury from October to September 2022. 53 eligible adolescents were selected purposefully. The participants were randomly divided into two groups: a family therapy counseling (n=27 people) and a control group (n=26 people). The experimental group received sixteen 75-min counseling sessions held weekly, while the control group underwent no interventions. Both groups received post-test evaluations following these sessions. Then, the Deliberate Self-Harm Inventory and Values in Action Inventory of Strengths scale (VIA-Youth) pre- and post-intervention were implemented. SPSS 24 used a multivariate analysis of the covariance (MANCOVA) version. Results: The effect of family therapy on the self-injury variable was significant (F=32.61; Pvalue&lt;0.001). Also, the effect of family therapy on character strengths (F=24.81; Pvalue&lt;0.001) was confirmed. In addition, it related the largest effect size to the self-injury variable (0.648), which shows that 64% of the total variances of the experimental and control groups result from a family therapy approach. Conclusions: The results showed that family therapy could effectively improve character strengths and reduce self-injurious behaviors in adolescents aged 13-16. Keywords: Family therapy, Adolescent, Self-harm, Character strengths