Related papers: Semiregularity and obstructions of complete inters…

Using Quillen-Lurie deformation theory formalism we develop an obstruction theory for studying the stable $\infty$-category of modules over a given geometric $\infty$-stack. The obstruction theory studies the problem of lifting compact…

Let $X$ be a Deligne-Mumford stack locally of finite type over an algebraically closed field $k$ of characteristic zero. We show that the intrinsic normal cone $C_X$ of $X$ is supported in the subcone $\mathbb{V}(\Omega_X[-1])$…

We characterize the graphs $G$ for which their toric ideals $I_G$ are complete intersections. In particular we prove that for a connected graph $G$ such that $I_G$ is complete intersection all of its blocks are bipartite except of at most…

Let $\overline G$ be the wonderful compactification of a simple affine algebraic group $G$ of adjoint type defined over $\mathbb C.$ Let ${\overline T}\subset \overline G$ be the closure of a maximal torus $T\subset G.$ We prove that the…

In this paper, we study relative deformations of maps into a family of K\"ahler manifolds whose images are divisors. We show that if the map satisfies a condition called semiregularity, then it allows relative deformations if and only if…

Let $X$ be a smooth irreducible projective variety of dimension at least 2 over an algebraically closed field of characteristic 0 in the projective space ${\mathbb{P}}^n$. Bertini's Theorem states that a general hyperplane $H$ intersects…

Let D be a smoothly bounded domain in a complex vector space of dimension n. Suppose that D has a smooth defining function, such that the sum of any q eigenvalues of its complex Hessian are non-negative on the closure of D. We show that…

In this paper, we prove the smoothness of the functors of locally trivial deformations, flat deformations and log smooth deformations for irreducible type II degeneration of complex abelian surfaces.

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…

In this paper we survey some results on the symmetric semi-perfect obstruction theory on a Deligne-Mumford stack $X$ constructed by Chang-Li, and Behrend's theorem equating the weighted Euler characteristic of $X$ and the virtual count of…

We study moduli of semistable twisted sheaves on smooth proper morphisms of algebraic spaces. In the case of a relative curve or surface, we prove results on the structure of these spaces. For curves, they are essentially isomorphic to…

Let $K$ be a local field of residue characteristic $p$. Let $C$ be a curve over $K$ whose minimal proper regular model has totally degenerate semi-stable reduction. Under certain hypotheses, we compute the prime-to-$p$ rational torsion…

Given a semistable degeneration with a simple normal crossings central fiber, Abramovich-Chen-Gross-Siebert [3] proved a degeneration formula that relates the moduli spaces of stable maps in smooth fibers to certain moduli spaces of…

We complete the remaining cases of the conjecture predicting existence of infinitely many rational curves on K3 surfaces in characteristic zero, prove almost all cases in positive characteristic and improve the proofs of the previously…

We show how to find a complete set of necessary and sufficient conditions that solve the fixed-parameter local congruence problem of immersions in $G$-spaces, whether homogeneous or not, provided that a certain $k^{\rm th}$ order jet bundle…

The categories with noninvertible morphisms are studied analogously to the semisupermanifolds with noninvertible transition functions. The concepts of regular n-cycles, obstruction and the regularization procedure are introduced and…

In this paper we introduce a model theoretic construction for the theories of uniform layered domains and semifields introduced in the paper of Izhakian, Knebusch and Rowen. We prove that, for a given layering semiring L, the theory of…

We give the construction of a class of multiple locally complete intersection structures on a smooth algebraic variety as support. This class contains the structures defined locally by equations of the form $x^n=0$, $y^2=0$, $z=0, >...,…

We continue the development of the infinitesimal deformation theory of pasting diagrams of k-linear categories begun in Yetter, D.N. "On Deformations of Pasting Diagrams", Theory and Applications of Categories 22 (2009) 24-53. In that…

This paper gives a uniform, self-contained and direct approach to a variety of obstruction-theoretic problems on manifolds of dimension 7 and 6. We give necessary and sufficient cohomological criteria for the existence of various…