相关论文: On secant defective varieties, in particular of di…
Beauville asked if a compact K\"ahler manifold with split tangent bundle has a universal covering that is a product of manifolds. We use Mori theory and elementary results about holomorphic foliations to study this problem for projective…
We show that a four-dimensional complete gradient shrinking Ricci soliton with positive isotropic curvature is either a quotient of S^4 or a quotient of S^3 cross R. This gives a clean classification result removing the earlier additional…
We give an introduction to the theory of wonderful G-varieties, with many examples when G is simple of type F4. We present results and open problems about these varieties: on their classification, on their isotropy groups, on morphisms…
We introduce the concise secant varieties, which are, informally speaking, modular partial desingularisations of secant varieties to Segre embeddings. More precisely, they are projective and birational to the abstract secant varieties, yet…
Let $X_{m,n}$ be the Segre-Veronese variety $\mathbb{P}^m \times \mathbb{P}^n$ embedded by the morphism given by $\mathcal{O}(1,2)$. In this paper, we provide two functions $\underline{s}(m,n)\le \bar{s}(m,n)$ such that the…
The $k$-secant degree is studied with a combinatorial approach. A planar toric degeneration of any projective toric surface $X$ corresponds to a regular unimodular triangulation $D$ of the polytope defining $X$. If the secant ideal of the…
Tropical geometry yields good lower bounds, in terms of certain combinatorial-polyhedral optimisation problems, on the dimensions of secant varieties. In particular, it gives an attractive pictorial proof of the theorem of Hirschowitz that…
We study linear series on a general curve of genus g, whose images are exceptional with respect to their secant planes. Each such exceptional secant plane is algebraically encoded by an included linear series, whose number of base points…
We generalise a construction of Landsberg, which associates certain Clifford algebra representations to Severi varieties. We thus obtain a new proof of Russo's Divisibility Property for LQEL varieties.
Given a variety $Y$ with a rectangular Lefschetz decomposition of its derived category, we consider a degree $n$ cyclic cover $X \to Y$ ramified over a divisor $Z \subset Y$. We construct semiorthogonal decompositions of $\mathrm{D^b}(X)$…
To complete the classification theory and the structure theory of varieties of almost minimal degree, that is of non-degenerate irreducible projective varieties whose degree exceeds the codimension by precisely 2, a natural approach is to…
We describe a computational proof that the fifth secant variety of the Segre product of five copies of the projective line is a codimension 2 complete intersection of equations of degree 6 and 16. Our computations rely on pseudo-randomness,…
For an irreducible projective variety X, we study the family of h-planes contained in the secant variety Sec_k(X), for 0<h<k. These families have an expected dimension and we study varieties for which the expected dimension is not attained;…
Restricted secant varieties of Grassmannians are constructed from sums of points corresponding to $k$-planes with the restriction that their intersection has a prescribed dimension. We study dimensions of restricted secant of Grassmannians…
We give a complete answer for the existence of Kawamata type semiorthogonal decompositions of derived categories of nodal del Pezzo threefolds. More precisely, we show that nodal del Pezzo threefolds of degree $1\leq d \leq 4$ have no…
We consider semi-orthogonal decompositions of derived categories for 3-dimensional projective varieties in the case when the varieties have ordinary double points.
We present existence results for certain singular 2-dimensional foliations on 4-manifolds. The singularities can be chosen to be simple, e.g. the same as those that appear in Lefschetz pencils. There seems to be a wealth of such creatures…
Consider a smooth projective curve and a given embedding into projective space via a sufficiently positive line bundle. We can form the secant variety of $k$-planes through the curve. These are singular varieties, with each secant variety…
We continue the study, begun by the second author in math.AG/0701889, of secant defective manifolds having "simple entry loci". We prove that such manifolds are rational and describe them in terms of tangential projections. Using also our…
In this paper, we show that the secant variety to a smooth projective variety embedded by a sufficiently positive line bundle is normal. As an application, we deduce that the secant variety to a general canonical curve of genus at least 7…