Criteria for complete intersections

We show that a refined version of Golyshev's canonical strip hypothesis does hold for the Hilbert polynomials of complete intersections in rational homogeneous spaces.

We prove that the derived category of a smooth complete intersection variety is equivalent to a full subcategory of the derived category of a smooth projective Fano variety. This enables us to define some new invariants of smooth projective varieties and raise many interesting questions.

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.

Using the connections among almost complete intersection schemes, arithmetically Gorenstein schemes and schemes that are union of complete intersections we give a structure theorem for arithmetically Cohen-Macaulay union of two complete intersections of codimension $2.$ We apply the results for computing Hilbert functions and graded Betti numbers for such schemes.

Let I = (F_1,...,F_r) be a homogeneous ideal of R = k[x_0,...,x_n] generated by a regular sequence of type (d_1,...,d_r). We give an elementary proof for an explicit description of the graded Betti numbers of I^s for any s \geq 1. These numbers depend only upon the type and s. We then use this description to: (1) write H_{R/I^s}, the Hilbert function of R/I^s, in terms of H_{R/I}; (2) verify that the k-algebra R/I^s satisfies a conjecture of Herzog-Huneke-Srinivasan; and (3) ...

Let X\subsetneq\mathbb{P}_{\mathbb{C}}^{N} be an n-dimensional nondegenerate smooth projective variety containing an m-dimensional subvariety Y. Assume that either m>\frac{n}{2} and X is a complete intersection or that m\geq\frac{N}{2}, we show deg(X)|deg(Y) and codim_{span(Y)}Y\geq codim_{\mathbb{P}^{N}}X, where span(Y) is the linear span of Y. These bounds are sharp. As an application, we classify smooth projective n-dimensional quadratic varieties swept out by m\geq[\frac{...

We prove that Fano complete intersections in projective spaces satisfy Conjecture $\mathcal O$ proposed by Galkin-Golyshev-Iritani.

We study the geometry of spaces of planes on smooth complete intersections of three quadrics, with a view toward rationality questions.

We study the minimal degrees and gonalities of curves on complete intersections. We prove a classical conjecture which asserts that the degree of any curve on a general complete intersection $X \subseteq \mathbb{P}^N$ cut out by polynomials of large degrees is bounded from below by the degree of $X$. As an application, we verify a conjecture of Bastianelli--De Poi--Ein--Lazarsfeld--Ullery on measures of irrationality for complete intersections.

In this paper we study the cohomology of tensor products of symmetric powers of the cotangent bundle of complete intersection varieties in projective space. We provide an explicit description of some of those cohomology groups in terms of the equations defining the complete intersection. We give several applications. First we prove a non-vanishing result, then we give a new example illustrating the fact that the dimension of the space of holomorphic symmetric differential for...