Hajime Kaji | Searcharxiv

arXiv2017

Higher Gauss Maps of Veronese Varieties---a generalization of Boole's formula and degree bounds for higher Gauss map images---

The image of the higher Gauss map for a projective variety is discussed. The notion of higher Gauss maps here was introduced by Fyodor L. Zak as a generalization of both ordinary Gauss maps and conormal maps. The main result is a closed formula for the degree of those images of Veronese varieties. This yields a generalization of a classical formula by George Boole on the degree of the dual varieties of Veronese varieties in 1844. As an application of our formula, degree bounds for higher Gauss map images of Veronese varieties are given.

math.AG↗

arXiv2015

Degree Formulae for Grassmann Bundles, II

Let $X$ be a non-singular quasi-projective variety over a field, and let $\mathcal E$ be a vector bundle over $X$. Let $\mathbb G_X({d}, \mathcal E)$ be the Grassmann bundle of $\mathcal E$ over $X$ parametrizing corank $d$ subbundles of $\mathcal E$ with projection $π: {\mathbb G_X({d}, \mathcal E)} \to X$, and let $ \mathcal Q \gets π^*\mathcal E$ be the universal quotient bundle of rank $d$. In this article, a closed formula for $π_{*}\operatorname{ch} (\det \mathcal Q)$, the push-forward of the Chern character of the Plücker line bundle $\det \mathcal Q$ by $π$ is given in terms of the Segre classes of $\mathcal E$. Our formula yields a degree formula for $\mathbb G_X({d}, \mathcal E)$ with respect to $\det \mathcal Q$ when $X$ is projective and $\wedge ^d \mathcal E$ is very ample. To prove the formula above, a push-forward formula in the Chow rings from a partial flag bundle of $\mathcal E$ to $X$ is given.

math.AG↗