[Paper Review] Normal class and normal lines of algebraic hypersurfaces

Alfrederic Josse, Françoise Pène|arXiv (Cornell University)|Feb 28, 2014

Algebraic Geometry and Number Theory11 references3 citations

TL;DR

This paper introduces the normal class of an algebraic hypersurface in complexified projective space, defined as the number of normal lines through a generic point. Using normal polars and Schubert calculus, it derives a general formula for the normal class of smooth hypersurfaces: $ c_{\nu}(\mathcal{Z}) = d_{\mathcal{Z}} \sum_{k=0}^{n-1} (d_{\mathcal{Z}}-1)^k $, with explicit values for degrees and dimensions, and extends it to singular surfaces via intersection multiplicities.

ABSTRACT

We are interested in the normal class of an algebraic hypersurface Z of the complex projective space P^n, that is the number of normal lines to Z passing through a generic point of P^n. Thanks to the notion of normal polar, we state a formula for the normal class valid for a general hypersurface Z of P^n. We give a generic result and we illustrate our formula with examples in P^n. We define the orthogonal indidence variety and compute the Schubert class of the variety of projective normal lines to a surface of P^3 in the Show ring of G(1,3). We complete our work with a generalization of Salmon's formula for the normal class of a Plucker curve to any planar curve with any kind of singularity.

Motivation & Objective

• To define and compute the normal class of an algebraic hypersurface in complexified projective space $\mathbb{P}^n$, i.e., the number of normal lines through a generic point.
• To generalize Salmon’s formula for Plücker curves to plane curves with arbitrary singularities.
• To study the geometry of projective normal lines using the orthogonal incidence variety and Schubert calculus in the Grassmannian $\mathbb{G}(1,n)$.
• To establish a formula for the normal class of hypersurfaces with isolated singularities, accounting for base points and intersection multiplicities.

Proposed method

• Introduce the notion of normal polars $\mathcal{P}_{A,\mathcal{Z}}$ as the locus of points $m$ such that $A$ lies on the normal line $\mathcal{N}_m(\mathcal{Z})$.
• Define a regular map $\alpha_{\mathcal{Z}}: \mathbb{P}^n \setminus \mathcal{B}^{(0)}_{\mathcal{Z}} \to \mathbb{P}^{\frac{n(n+1)}{2}-1}$ sending $m$ to its normal line $\mathcal{N}_m(\mathcal{Z})$.
• Characterize the base locus $\mathcal{B}_{\mathcal{Z}} = \mathcal{B}^{(0)}_{\mathcal{Z}} \cap \mathcal{Z}$ as the union of singular points, tangency points with $\mathcal{H}^\infty$, and tangency points of $\mathcal{Z}_\infty$ with the umbilic $\mathcal{U}_\infty$.
• Use Schubert calculus in $\mathbb{G}(1,n)$ to compute the Schubert class $\mathfrak{n}_{\mathcal{Z}}$ of the variety of normal lines.
• Apply intersection theory to compute the normal class as $ c_{\nu}(\mathcal{Z}) = d_{\mathcal{Z}} \sum_{k=0}^{n-1} (d_{\mathcal{Z}}-1)^k - \sum_{P \in \mathcal{B}_{\mathcal{Z}}} i_P(\mathcal{Z}, \mathcal{P}_{A,\mathcal{Z}}) $ for generic $A$.
• Establish projective orthogonality via a bilinear form on $\mathbf{V} = (\mathbf{E}_n \oplus \mathbb{R}) \otimes \mathbb{C}$, defining orthogonality between lines and hyperplanes in $\mathbb{P}^n$.

Experimental results

Research questions

• RQ1 What is the number of normal lines to a smooth hypersurface $\mathcal{Z} \subset \mathbb{P}^n$ passing through a generic point in $\mathbb{P}^n$?
• RQ2 How does the normal class change when $\mathcal{Z}$ has isolated singularities or tangency with the hyperplane at infinity?
• RQ3 Can Salmon’s formula for the normal class of a Plücker curve be generalized to singular plane curves?
• RQ4 What is the Schubert class of the variety of projective normal lines to a surface in $\mathbb{P}^3$?
• RQ5 How does the non-invariance under $PGL(n,\mathbb{C})$ affect the study of normal lines compared to tangent hyperplanes?

Key findings

• For a smooth hypersurface $\mathcal{Z} \subset \mathbb{P}^n$ of degree $d_{\mathcal{Z}} \geq 2$ with no tangency to $\mathcal{H}^\infty$ and transverse tangency at infinity, the normal class is $ c_{\nu}(\mathcal{Z}) = d_{\mathcal{Z}} \sum_{k=0}^{n-1} (d_{\mathcal{Z}}-1)^k $.
• For a quadric ($d_{\mathcal{Z}} = 2$), the normal class is $c_{\nu}(\mathcal{Z}) = n$, and for $n=2$, it is $d_{\mathcal{Z}}$.
• In $\mathbb{P}^3$, the normal class is $d_{\mathcal{Z}}^3 - d_{\mathcal{Z}}^2 + d_{\mathcal{Z}}$, and in $\mathbb{P}^4$, it is $d_{\mathcal{Z}}^4 - 2d_{\mathcal{Z}}^3 + 2d_{\mathcal{Z}}^2$.
• For $n=5$, the normal class is $d_{\mathcal{Z}}^5 - 3d_{\mathcal{Z}}^4 + 4d_{\mathcal{Z}}^3 - 2d_{\mathcal{Z}}^2 + d_{\mathcal{Z}}$.
• The normal class of a hyperplane $\mathcal{H} \subset \mathbb{P}^n$ (other than $\mathcal{H}^\infty$) is $c_{\nu}(\mathcal{H}) = 1$.
• The general formula for the normal class of a hypersurface with isolated singularities includes a correction term: $ c_{\nu}(\mathcal{Z}) = d_{\mathcal{Z}} \sum_{k=0}^{n-1} (d_{\mathcal{Z}}-1)^k - \sum_{P \in \mathcal{B}_{\mathcal{Z}}} i_P(\mathcal{Z}, \mathcal{P}_{A,\mathcal{Z}}) $, where $i_P$ is the intersection multiplicity at base points.