## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Singularities in Geometry and Topology 2011, V. Blanlœil and O. Saeki, eds. (Tokyo: Mathematical Society of Japan, 2015), 1 - 11

### Smooth double subvarieties on singular varieties. II

Maria del Rosario Gonzalez-Dorrego

#### Abstract

Let k be an algebraically closed field of characteristic 0. We give a brief survey on multiplicity-2 structures on varieties. Let $Z$ be a reduced irreducible nonsingular $(n-1)$-dimensional variety such that $2Z = X \cap F$, where $X$ is a normal $n$-fold with canonical singularities, $F$ is an $(N-1)$-fold in $\mathbb{P}^N$, such that $Z \cap \mathrm{Sing}(X) \neq \emptyset$. Assume that $\mathrm{Sing}(X)$ is equidimensional and $\mathrm{codim}_X(\mathrm{Sing}(X)) = 3$. We study the singularities of $X$ through which $Z$ passes. We also consider Fano cones. We discuss the construction of some vector bundles and the resolution property of a variety.

#### Article information

**Dates**

Received: 4 May 2012

Revised: 14 January 2013

First available in Project Euclid:
19 October 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1539916276

**Digital Object Identifier**

doi:10.2969/aspm/06610001

**Mathematical Reviews number (MathSciNet)**

MR3382039

**Zentralblatt MATH identifier**

1360.14012

**Subjects**

Primary: 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx]

Secondary: 14E15: Global theory and resolution of singularities [See also 14B05, 32S20, 32S45] 32S25: Surface and hypersurface singularities [See also 14J17] 14J17: Singularities [See also 14B05, 14E15] 14J30: $3$-folds [See also 32Q25] 14J35: $4$-folds 14J40: $n$-folds ($n > 4$) 14J70: Hypersurfaces

**Keywords**

$n$-fold singularity intersection

#### Citation

Gonzalez-Dorrego, Maria del Rosario. Smooth double subvarieties on singular varieties. II. Singularities in Geometry and Topology 2011, 1--11, Mathematical Society of Japan, Tokyo, Japan, 2015. doi:10.2969/aspm/06610001. https://projecteuclid.org/euclid.aspm/1539916276