Related papers: Derived geometry of the first formal neighborhood …

The quantum cohomology algebra of a projective manifold X is the cohomology H(X,Q) endowed with a different algebra structure, which takes into account the geometry of rational curves in X. We show that this algebra takes a remarkably…

We give a short proof of the following result: Let $X$ be a complex surface of general type. If the canonical divisor of the minimal model of $X$ has selfintersection $= 1$, then $X$ is not diffeomorphic to a rational surface. Our proof is…

We examine the extent to which a smooth minimal complex projective surface X is determined by its derived category of coherent sheaves D(X). To do this we find, for each such surface X, the set of surfaces Y for which there exists a…

A discretisation scheme that preserves topological features of a physical problem is extended so that differential geometric structures can be approximated in a consistent way thus giving access to the study of physical systems which are…

Let M be a totally orientable graph manifold with characteristic submanifold T and let M = V cup_S W be a Heegaard splitting. We prove that S is standard. In particular, S is the amalgamation of strongly irreducible Heegaard splittings. The…

The need to compute the intersections between a line and a high-order curve or surface arises in a large number of finite element applications. Such intersection problems are easy to formulate but hard to solve robustly. We introduce a…

Let $X$ be a smooth simply connected closed 4-manifold with definite intersection form. We show that any automorphism of the intersection form of $X$ is realized by a diffeomorphism of $X \mathbin{\#} S^2 \times S^2$. This extends and…

Let X be a smooth rational surface. We calculate a DG quiver of a full exceptional collection of line bundles on X obtained by an augmentation from a strong exceptional collection on the minimal model of X. In particular, we calculate…

Let $i:X\hookrightarrow Y$ be a closed embedding of smooth algebraic varieties. Denote by $N$ the normal bundle of $X$ in $Y$. We describe the construction of two Lie-type structures on the shifted bundle $N[-1]$ which encode the…

In this paper we produce the first known formula for cohomologies of the derived tensor products of structure sheaves of subschemes in the case where the intersection of the subschemes is not a local complete intersection. The case covered…

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 $\mathrm{D^b}(X)$…

For an oriented surface $S$, the singular set of a fold map $f:S\rightarrow \mathbb{R}^2$ is a collection of smooth curves, also known as fold singularities. We construct a sharp lower bound on the number of self-intersections of such fold…

An intersection digraph is a digraph where every vertex $v$ is represented by an ordered pair $(S_v, T_v)$ of sets such that there is an edge from $v$ to $w$ if and only if $S_v$ and $T_w$ intersect. An intersection digraph is reflexive if…

We prove that every smoothly embedded surface in a 4--manifold can be isotoped to be in bridge position with respect to a given trisection of the ambient 4--manifold; that is, after isotopy, the surface meets components of the trisection in…

We continue the explorations of derived \canal geometry started in [DAG-IX] and in http://arxiv.org/abs/1506.09042. We describe the category of $\mathcal O_X$-modules over a derived complex analytic space $X$ as the stabilization of a…

We study a class of complex polynomial equations on a finite graph with a view to understanding how holistic phenomena emerge from combinatorial structure. Particular solutions arise from orthogonal projections of regular polytopes,…

We show that any tetragonal Gorenstein integral curve is a complete intersection in its respective $3$-fold rational normal scroll S, implying that the normal sheaf on $C$ embedded in S, and in $\mathbb{P}^{g-1}$ as well, is unstable for…

Let $X$ be an algebraic stack with quasi-affine diagonal of finite type over a field $k$ of characteristic $0$. We extend the well-known equivalence $\mathsf{D}^+(\mathsf{QCoh}(X)) \simeq \mathsf{D}_{\mathrm{qc}}^+(X)$ to unbounded derived…

Computing a quasi-developable strip surface bounded by design curves finds wide industrial applications. Existing methods compute discrete surfaces composed of developable lines connecting sampling points on input curves which are not…

Consider the scheme parametrizing non-constant morphisms from a fixed projective curve to a projective surface. There is a rational map between this scheme and the Chow variety of $1$-cycles on the surface. We prove that, if the curve is…