Fano Schemes of Complete Intersections in Toric Varieties

We get sharp degree bound for generic smoothness and connectedness of the space of conics in low degree complete intersections which generalizes the old work about Fano scheme of lines on Hypersurfaces.

We study the Picard rank of smooth toric Fano varieties possessing families of minimal rational curves of given degree. We discuss variants of a conjecture of Chen-Fu-Hwang and prove a version of their statement that recovers the original conjecture in sufficiently high dimension. We also prove new cases of the original conjecture for high degrees in all dimensions. Our main tools come from toric Mori theory and the combinatorics of Fano polytopes.

Given a set $\mathcal A = \{a_1,\ldots,a_n\} \subset \mathbb{N}^m$ of nonzero vectors defining a simplicial toric ideal $I_{\mathcal A} \subset k[x_1,...,x_n]$, where $k$ is an arbitrary field, we provide an algorithm for checking whether $I_{\mathcal A}$ is a complete intersection. This algorithm does not require the explicit computation of a minimal set of generators of $I_{\mathcal A}$. The algorithm is based on the application of some new results concerning toric ideals t...

We present a class of toric varieties $V$ which, over any algebraically closed field of characteristic zero, are defined by codim $V$+1 binomial equations.

We provide enumerative formulas for the degrees of varieties parameterizing hypersurfaces and complete intersections which contain pro-jective subspaces and conics. Besides, we find all cases where the Fano scheme of the general complete intersection is irregular of dimension at least 2, and for the Fano surfaces we deduce formulas for their holomorphic Euler characteristic.

For a smooth complete intersection X, we consider a general fiber \mathbb{F} of the evaluation map ev of Kontsevich moduli space \bar{M}_{0,m}(X,m)\rightarrow X^m and the forgetful functor F : \mathbb{F} \rightarrow \bar{M}_{0,m}. We prove that a general fiber of the map $F$ is a smooth complete intersection if $X$ is of low degree. The result sheds some light on the arithmetic and geometry of Fano complete intersections.

This article gives an overview of toric Fano and toric weak Fano varieties associated to graphs and building sets. We also study some properties of such toric Fano varieties and discuss related topics.

Let $X_{\Sigma}$ be a smooth complete toric variety defined by a fan $\Sigma$ and let $V=V(I)$ be a subscheme of $X_{\Sigma}$ defined by an ideal $I$ homogeneous with respect to the grading on the total coordinate ring of $X_{\Sigma}$. We show a new expression for the Segre class $s(V,X_{\Sigma})$ in terms of the projective degrees of a rational map specified by the generators of $I$ when each generator corresponds to a numerically effective (nef) divisor. Restricting to the ...

The main purpose of this paper is to give a simple and non-combinatorial proof of the toric Mori theory. Here, the toric Mori theory means the (log) Minimal Model Program (MMP, for short) for toric varieties. We minimize the arguments on fans and their decompositions. We recommend this paper to the following people: (A) those who are uncomfortable with manipulating fans and their decompositions, (B) those who are familiar with toric geometry but not with the MMP. People i...

In this paper we propose and prove an explicit formula for computing the degree of Fano schemes of linear subspaces on general hypersurfaces. The method used here is based on the localization theorem and Bott's residue formula in equivariant intersection theory.