Related papers: Sarkisov Links with centres space curves on smooth…

We show that 3-fold terminal flips and divisorial contractions to a curve may be factored by a sequence of weighted blow-ups, flops, blow-downs to a locally complete intersection curve in a smooth 3-fold or divisorial contractions to a…

In this paper, an update on the classification of smooth weak Fano threefolds with Picard number two and small anti-canonical maps is given. Geometric constructions are provided for previously open numerical cases by blowing up certain…

We consider links that are alternating on surfaces embedded in a compact 3-manifold. We show that under mild restrictions, the complement of the link decomposes into simpler pieces, generalising the polyhedral decomposition of alternating…

We study the generalization of quasipositive links from the three-sphere to arbitrary closed, orientable three-manifolds. Our main result shows that the boundary of any smooth, properly embedded complex curve in a Stein domain is a…

We use the cut and paste relation $[Y^{[2]}] = [Y][\mathbb{P}^m] + \mathbb{L}^2 [F(Y)]$ in $K_0(\text{Var}_k)$ of Galkin--Shinder for cubic hypersurfaces arising from projective geometry to characterize cubic hypersurfaces of sufficiently…

In this article, we study the K-moduli space of Fano threefolds obtained by blowing up $\mathbb{P}^3$ along $(2,3)$-complete intersection curves. This K-moduli space is a two-step birational modification of the GIT moduli space of…

We give a classification and a construction of all smooth $(n-1)$-dimensional varieties of lines in ${\bf P}\sp n$ verifying that all their lines meet a curve. This also gives a complete classification of $(n-1)$-scrolls over a curve…

We give an explicit presentation with generators and relations of the quantum cohomology ring of the blow-up of a projective space along a linear subspace.

We prove that for any closed surface of genus at least four, and any punctured surface of genus at least two, the space of ending laminations is connected. A theorem of E. Klarreich implies that this space is homeomorphic to the Gromov…

We describe the closed cone of moving curves of smooth Fano three- and fourfolds by giving finitely many equations that cut out the cone. The equations are induced by the exceptional divisors of divisorial contractions and by nef divisors…

We consider the subspace of the homology of a covering space spanned by lifts of simple closed curves. Our main result is the existence of unbranched covers of surfaces where this is a proper subspace. More generally, for a fixed finite…

A classical result of Bondal-Orlov states that a standard flip in birational geometry gives rise to a fully faithful functor between derived categories of coherent sheaves. We complete their embedding into a semiorthogonal decomposition by…

We discuss links in thickened surfaces. We define the Khovanov-Lipshitz-Sarkar stable homotopy type and the Steenrod square for the homotopical Khovanov homology of links in thickened surfaces with genus$>1$. A surface means a closed…

In this notes we study complex projective plane curves whose graded module of Jacobian syzygies is generated by its minimal degree component. Examples of such curves include the smooth curves as well as the maximal Tjurina curves. However,…

We consider the problem to determine which blow-ups along subvarieties in products of two projective spaces are log Fano. By describing the nef cones of such blow-ups with special centers, we give a partial classification result. For each…

We study the notion of symplectic scalar curvature on the supermanifold over an ordinary Fedosov manifold whose structural sheaf is that of differential forms. In this purely geometric context, we introduce two families of odd super-Fedosov…

This paper examines the relationship between the knotting of an embedded surface in $\R^3$ and the knotting of its fold curves, formed by the singular set of projection to a plane. The first result shows that every surface, no matter how…

Using ideas from the theory of tropical curves and degeneration, we prove that any Fano hypersurface (and more generally Fano complete intersections) is swept by at most quadratic rational curves.

We prove that each divisorial contraction to a curve between terminal threefolds is a weighted blow-up under a suitable embedding. Moreover, we give a classification of the weighted blow-ups assuming that the curve is smooth.

We show that bounded self-contracted curves are rectifiable in metric spaces with weak lower curvature bound in a sense we introduce in this article. This class of spaces is wide and includes, for example, finite-dimensional Alexandrov…