## Asian Journal of Mathematics

- Asian J. Math.
- Volume 18, Number 3 (2014), 525-544.

### Symmetry defect of algebraic varieties

S. Janeczko, Z. Jelonek, and M. A. S. Ruas

#### Abstract

Let $X, Y \subset k^m(k = \mathbb{R},\mathbb{C})$ be smooth manifolds. We investigate the central symmetry of the configuration of $X$ and $Y$. For $p \in k^m$ we introduce a number $\mu(p)$ of pairs of
points $x \in X$ and $y \in Y$ such that $p$ is the center of the interval $\overline{xy}$. We show that if $X, Y$ (including the case $X = Y$ ) are algebraic manifolds in a general position, then there is a closed
(semi-algebraic) set $B \subset k^m$, called *symmetry defect* set of the $X$ and $Y$ configuration, such that the function $\mu$ is locally constant and not identically zero outside $B$. If $k = \mathbb{C}$, we
estimate the number $\mu$ (in fact we compute it in many cases) and show that the symmetry defect is an algebraic hypersurface and consequently the function $\mu$ is constant and positive outside $B$. We also
show that in the generic case the topological type of the symmetry defect set of a plane curve is constant, i.e. the symmetry defect sets for two generic curves of the same degree are homeomorphic (by the same
method we can prove similar statement for any irreducible family of smooth varieties $Z^n \subset \mathbb{C}^{2n}$). Moreover, for $k = \mathbb{R}$, we estimate the number of connected components of the
set $U = k^m \backslash B$. In the last section we give an algorithm to compute the symmetry defect set for complex smooth affine varieties in general position.

#### Article information

**Source**

Asian J. Math., Volume 18, Number 3 (2014), 525-544.

**Dates**

First available in Project Euclid: 8 September 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.ajm/1410186670

**Mathematical Reviews number (MathSciNet)**

MR3257839

**Zentralblatt MATH identifier**

1343.14050

**Subjects**

Primary: 14D06: Fibrations, degenerations 14Q20: Effectivity, complexity

**Keywords**

Polynomial mapping fibration bifurcation points center symmetry set Wigner caustic

#### Citation

Janeczko, S.; Jelonek, Z.; Ruas, M. A. S. Symmetry defect of algebraic varieties. Asian J. Math. 18 (2014), no. 3, 525--544. https://projecteuclid.org/euclid.ajm/1410186670