Paltin Ionescu

Generalized adjunction and applications

The linear system |K + C| ‘adjoint’ to a curve C on a projective surface was studied by the classical Italian geometers. The adjoint system to a hyperplane section H of smooth projective surface was investigated systematically, in modern terms, by Sommese [22] and Van de Ven [26]. The map associated to the linear system | K + ( r −1) H |, where H is a hyperplane section of a smooth variety of arbitrary dimension r , was used to classify submanifolds of ℙ n with ‘small invariants’ (e.g. degree, sectional genus, etc.); see [10]. On the other hand, Sommese [ 23, 24, 25 ] studied adjoint systems to a smooth ample divisor H on a smooth threefold X and obtained, as applications, many interesting results about the pair ( X, H ). As noticed independently by several authors (see e.g. [17], [4], [11]) the appearance of Moris deep contribution [20] (see also [21]) put the subject of adjunction in a new perspective. Accordingly, the present paper–which relies heavily on Moris results and on the contraction theorem due to Kawamata-Shokurov (see [14])–contains a systematical study of various adjoint systems to an ample (possibly non-effective) divisor on a manifold of arbitrary dimension. More precisely, the main result (which is contained in Section 1) gives the precise description of polarized pairs ( X, H ), where X is a complex projective mani–fold of dimension r and H an ample divisor on it (not necessarily effective), such that K x + iH is not semiample (respectively ample) for 1 ≤ i = r + 1, r , r − 1, r − 2 (respectively i = r + 1, r , r − 1).

Conic-connected manifolds

Abstract We study a particular class of rationally connected manifolds, , such that two general points x, x′ ∈ X may be joined by a conic contained in X. We prove that these manifolds are Fano, with b 2 ≦ 2. Moreover, a precise classification is obtained for b 2 = 2. Complete intersections of high dimension with respect to their multi-degree provide examples for the case b 2 = 1. The proof of the classification result uses a general characterization of rationality, in terms of suitable covering families of rational curves.

On Very Ample Vector Bundles on Curves

We study very ample vector bundles on curves. We first give numerical conditions for the existence of non-special such bundles. Then we prove the inequality \[ h^0(\det E)\ge h^0(E) + {\rm rank}(E)-2 \] over curves of genus at least two. We apply this to prove some special cases of a conjecture on scrolls of small codimension.

Manifolds covered by lines and the Hartshorne Conjecture for quadratic manifolds

Small codimensional embedded manifolds defined by equations of small degree are Fano and covered by lines. They are complete intersections exactly when the variety of lines through a general point is so and has the right codimension. This allows us to prove the Hartshorne Conjecture for manifolds defined by quadratic equations and to obtain the list of such Hartshorne manifolds.

RATIONALITY PROPERTIES OF MANIFOLDS CONTAINING QUASI-LINES

Let X be a complex, rationally connected, projective manifold. We show that X admits a modification that contains a quasi-line, i.e. a smooth rational curve whose normal bundle is a direct sum of copies of . For manifolds containing quasi-lines, a sufficient condition of rationality is exploited: there is a unique quasi-line from a given family passing through two general points. We define a numerical birational invariant, e(X), and prove that X is rational if and only if e(X)=1. If X is rational, there is a modification which is strongly-rational, i.e. contains an open subset isomorphic to an open subset of the projective space whose complement is at least 2-codimensional. We prove that strongly-rational varieties are stable under smooth, small deformations. The argument is based on a convenient characterization of these varieties. Finally, we relate the previous results and formal geometry. This relies on ẽ(X, Y), a numerical invariant of a given quasi-line Y that depends only on the formal completion . As applications we show various instances in which X is determined by . We also formulate a basic question about the birational invariance of ẽ(X, Y).

ON MANIFOLDS OF SMALL DEGREE

Let X ? Pn be a complex connected projective, non-degenerate, linearly normal manifold of degree d = n. The main result of this paper is a classification of such manifolds. As a by-product of the classification it follows that these manifolds are either rational or Fano. In particular, they are simply connected (hence regular) and of negative Kodaira dimension. Moreover, easy examples show that d = n is the best possible bound for such properties to hold true. The proof of our theorem makes essential use of the adjunction mapping and, in particular, the main result of [15] plays a crucial role in the argument.

On a Theorem of Van de Ven

A result by Van de Ven characterizes linear subspaces as the only closed submanifolds X ⊂ ℙ N for which the normal bundle exact sequence splits. We show that X is linear assuming only the splitting of the same exact sequence when restricted to some curve contained in X .

Remarks on defective Fano manifolds

This note continues our previous work on special secant defective (specifically, conic connected and local quadratic entry locus) and dual defective manifolds. These are now well understood, except for the prime Fano ones. Here we add a few remarks on this case, completing the results in our papers (Russo in Math Ann 344:597–617, 2009; Ionescu and Russo in Compos Math 144:949–962, 2008; Ionescu and Russo in J Reine Angew Math 644:145–157, 2010; Ionescu and Russo in Am J Math 135:349–360, 2013; Ionescu and Russo in Math Res Lett 21:1137–1154, 2014); see also the recent book (Russo, On the Geometry of Some Special Projective Varieties, Lecture Notes of the Unione Matematica Italiana, Springer, 2016).

Recognizing ℙn in Classical and Modern Setting

In classical projective algebraic geometry, ℙn was seen mainly as a linear subspace. The modern setting has produced in the last 40 years several remarkable abstract characterizations of projective space. We survey some interaction between these two points of view.