Related papers: Fano weighted complete intersections of large codi…

A variety is rationally connected if two general points can be joined by a rational curve. A higher version of this notion is rational simple connectedness, which requires suitable spaces of rational curves through two points to be…

The anticanonical complex generalizes the Fano polytope from toric geometry and has been used to study Fano varieties with torus action so far. We work out the case of complete intersections in toric varieties defined by non-degenerate…

We prove divisorial canonicity of Fano hypersurfaces and double spaces of general position with elementary singularities.

We find all K-stable smooth Fano threefolds in the family No. 2.22.

We study the geometry of the Fano schemes $\mathrm{\textbf{F}}_{k}(\mathrm{SD}_n^r)$ of the projective variety $\mathrm{SD}_n^r$ defined by the $r\times r$ minors of a symmetric $n\times n$ matrix filled with indeterminates. These schemes…

We prove divisorial canonicity of Fano double hypersurfaces of general position.

We show that Fano lattice polygons define a class of balanced quivers with interesting properties. The combinatorics of these quivers is related to singularities of the underlying toric Fano surface. This allows us to show that every Fano…

It is proved that the global log canonical threshold of a Zariski general Fano complete intersection of index 1 and codimension $k$ in ${\mathbb P}^{M+k}$ is equal to one, if $M\geqslant 2k+3$ and the maximum of the degrees of defining…

For smooth complete intersections in the projective spaces, we use the deformation invariance of Gromov-Witten invariants and results in classical invariant theory to study the symmetric reduction of the WDVV equation by the monodromy…

In this paper we describe all possible reduced complete intersection sets of points on Veronese surfaces. We formulate a conjecture for the general case of complete intersection subvarieties of any dimension and we prove it in the case of…

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

We complete the analysis on the birational rigidity of quasismooth Fano 3-fold deformation families appearing in the Graded Ring Database as a complete intersection. When such a deformation family $X$ has Fano index at least 2 and is…

We study the representation dimension of the class of algebras known as quantum complete intersections. For such an algebra, we show that the representation dimension is at most twice its codimension. Moreover, we show that the…

We classify compact K\"ahler manifolds with semi-positive holomorphic bisectional and big tangent bundles. We also classify compact complex surfaces with semi-positive tangent bundles and compact complex $3$-folds of the form $P(T^*X)$…

We prove a characterization of Fano type varieties.

For Fano varieties of various singularities such as canonical and terminal, we construct examples with large Fano index. By low-dimensional evidence, we conjecture that our examples have the largest Fano index for all dimensions.

A projective log variety (X, D) is called "a log Fano manifold" if X is smooth and if D is a reduced simple normal crossing divisor on X with -(K_X+D) ample. The n-dimensional log Fano manifolds (X, D) with nonzero D are classified in this…

We prove that a weak Fano manifold has unobstructed deformations. For a general variety, we investigate conditions under which a variety is necessarily obstructed.

We consider limit sets of some conformal iterated function systems, and introduce classes of subsets of the limit set, with the property that the classes are closed under countable intersections and all sets in the classes have large…

We extend our model for affine structures on toric Calabi-Yau hypersurfaces math.AG/0205321 to the case of complete intersections.