Related papers: Derivations of an effective divisor on the complex…

For every $g\geq 2$ and $n\geq g+1$ we exhibit infinitely many extremal effective divisors in $\overline{\mathcal{M}}_{g,n}$ coming from the strata of abelian differentials.

Let $X$ be a nonsingular variety defined over an algebraically closed field of characteristic $0$, and $D$ be a free divisor with Jacobian ideal of linear type. We compute the Chern class of the sheaf of logarithmic derivations along $D$…

Let $k$ be a commutative ring and $A$ a commutative $k$-algebra. Given a positive integer $m$, or $m=\infty$, we say that a $k$-linear derivation $\delta$ of $A$ is $m$-integrable if it extends up to a Hasse--Schmidt derivation…

This article concerns a question asked by M. V. Nori on homotopy of sections of Projective modules defined on the polynomial algebra over a smooth affine domain $R$. While this question has an affirmative answer, it is known that the…

We prove a strengthened form of a conjecture of Sun on a determinant attached to a binary quadratic form. Let $n>3$ and let $c,d\in\Z$. If $n$ is composite, then \[ \det\big[(i^2+cij+dj^2)^{n-2}\big]_{0\leq i,j\leq n-1}\equiv 0\pmod {n^2}…

In these notes we study hyperplane arrangements having at least one logarithmic derivation of degree two that is not a combination of degree one logarithmic derivations. It is well-known that if a hyperplane arrangement has a linear…

Let $X$ be a smooth projective variety. The Iitaka dimension of a divisor $D$ is an important invariant, but it does not only depend on the numerical class of $D$. However, there are several definitions of ``numerical Iitaka dimension'',…

We establish a shape-derivative formula for the Dirichlet-to-Neumann operator on a compact manifold. Then, we apply this formula to obtain the well-posedness in H 1 under a specific Rayleigh-Taylor condition to an equation describing cell…

We prove that for any finite-dimensional differential graded algebra with separable semisimple part the category of perfect modules is equivalent to a full subcategory of the category of perfect complexes on a smooth projective scheme with…

In this paper, we propose a definition of the moduli stack of stable relative ideal sheaves, and prove that it is a separated and proper Deligne-Mumford stack. It is the first part of the project of relative Donaldson-Thomas theory of ideal…

A complex hypersurface D in complex affine n-space C^n is a linear free divisor (LFD) if its module of logarithmic vector fields has a global basis of linear vector fields. We classify all LFDs for n at most 4. Analogous to Grothendieck's…

Let $X$ be a smooth projective rational surface, $D\subset X$ an effective anticanonical curve, $\beta$ a curve class on $X$ and $\mathfrak{d}=\sum w_iP_i$ an effective divisor on $D_{\mathrm{sm}}$. We consider the moduli space…

The free propagator for the scalar $\lambda \phi^4$--theory is calculated exactly up to the second derivative of a background field. Using this propagator I compute the one--loop effective action, which then contains all powers of the field…

Let $G$ be a connected reductive algebraic group $G$ over an algebraically closed field $k$ of prime characteristic $p$, and $\ggg=\Lie(G)$. In this paper, we study modular representations of the reductive Lie algebra $\ggg$ with…

We calculate the decomposition series of the D-module defined as the push-forward of a rank one linear system on the complement of a normal crossings hyperplane configuration and use data of a resolution of singularities to give a…

Suppose X is a projective variety, which needs not be smooth, and L an ample divisor on X. We show that there are integers c and b such that for any nonnegative integer p, L^d is normally generated and embeds X as a variety who defining…

Let $\mathcal{I}_{d,g,R}$ be the union of irreducible components of the Hilbert scheme whose general points parametrize smooth, irreducible, curves of degree $d$, genus $g$, which are non--degenerate in the projective space $\mathbb{P}^R$.…

We introduce a derived enhancement of the moduli space of sections defined by Chang-Li, and we compute its tangent complex. Special cases of this moduli space include stable maps and stable quasi-maps. As an application, we prove that…

A generic strictly semistable bundle of degree zero over a curve X has a reducible theta divisor, given by the sum of the theta divisors of the stable summands of the associated graded bundle. The converse is not true: Beauville and Raynaud…

Let A = (A,V) be a complex hyperplane arrangement and let L(A) denote its intersection lattice. The arrangement A is called supersolvable, provided its lattice L(A) is supersolvable, a notion due to Stanley. Jambu and Terao showed that…