Deformation of finite morphisms and smoothing of ropes

Smoothing of rational m-ropes

In a recent paper, Gallego, González and Purnaprajna showed that rational 3-ropes can be smoothed. We generalise their proof and obtain smoothability of rational $m$-ropes for $m \geq 3$.

An infinitesimal condition to deform a finite morphism to an embedding

In this article we give a sufficient condition for a morphism $φ$ from a smooth variety $X$ to projective space, finite onto a smooth image, to be deformed to an embedding. This result puts some theorems on deformation of morphisms of curves and surfaces such as $K3$ and general type, obtained by ad hoc methods, in a new, more conceptual light. One of the main interests of our result is to apply it to the construction of smooth varieties in projective space with given invariants. We illustrate this by using our result to construct canonically embedded surfaces with $c_1^2=3p_g-7$ and derive so…

Smoothable locally Cohen--Macaulay and non Cohen--Macaulay multiple structures on curves

In this article we show that a wide range of multiple structures on curves arise whenever a family of embeddings degenerates to a morphism $φ$ of degree $n$. One could expect to see, when an embedding degenerates to such a morphism, the appearance of a locally Cohen-Macaulay multiple structure of certain kind (a so-called rope of multiplicity $n$). We show that this expectation is naive and that both locally Cohen-Macaulay and non Cohen-Macaulay multiple structures might occur in this situation. In seeing this we find out that many multiple structures can be smoothed. When we specialize to the…

K3 carpets on minimal rational surfaces and their smoothings

In this article, we study K3 double structures on minimal rational surfaces $Y$. The results show there are infinitely many non-split abstract K3 double structures on $Y = \mathbb{F}_e$ parametrized by $\mathbb P^1$, countably many of which are projective. For $Y = \mathbb{P}^2$ there exist a unique non-split abstract K3 double structure which is non-projective (see Drézet's article in arXiv:2004.04921). We show that all projective K3 carpets can be smoothed to a smooth K3 surface. One of the byproducts of the proof shows that unless $Y$ is embedded as a variety of minimal degree, there are in…

具有简单正规交叉点的代数簇的投射光滑化

Projective smoothing of varieties with simple normal crossings

该研究提出基于态射形变理论与ribbons概形的新方法，证明投射空间中带两个Fano分支的简单正规交叉簇的存在性与光滑化，给出三维簇的显式构造及对应slc对的光滑化结果。

Log Smooth Deformation Theory

This paper gives a foundation of log smooth deformation theory. We study the infinitesimal liftings of log smooth morphisms and show that the log smooth deformation functor has a representable hull. This deformation theory gives, for example, the following two types of deformations: (1) relative deformations of a certain kind of a pair of an algebraic variety and a divisor of it, and (2) global smoothings of normal crossing varieties. The former is a generalization of the relative deformation theory introduced by Makio, and the latter coincides with the logarithmic deformation theory introduce…