退化判别式
Degenerating Discriminants
AI总结：
研究射影对偶簇和判别式在平坦退化下的行为，通过格罗贝纳退化等方法，描述退化特殊纤维情况并推广到格拉斯曼流形，还得到相关经典公式并联系混合判别式，展现对偶簇和判别式退化行为及应用。
AI中文摘要：
我们研究射影对偶簇和判别式在平坦退化下的行为。对于格罗贝纳退化，我们表明余法丛在对偶坐标上具有相反权重时允许格罗贝纳退化。利用惠特尼分层和萨巴赫公式，我们描述了这种退化特殊纤维的不可约分量和重数以及极限对偶超曲面的情况。然后将这些结果推广到格拉斯曼流形中的高阶相关超曲面。作为应用，我们得到了具有孤立奇点超曲面的经典公式，分析了一般完全交和互反线性空间的退化，并将该理论与参数化多项式系统的混合判别式联系起来。
英文摘要：
We study the behavior of projective dual varieties and discriminants under flat degenerations. For Gröbner degenerations, we show that the conormal variety admits a Gröbner degeneration with opposite weights on the dual coordinates. Using Whitney stratifications and Sabbah's formula, we describe the irreducible components and multiplicities of the special fiber of this degeneration, and hence of the limiting dual hypersurface. We then extend these results to higher associated hypersurfaces in Grassmannians. As applications, we recover classical formulas for hypersurfaces with isolated singularities, analyze degenerations of generic complete intersections and reciprocal linear spaces, and relate the theory to mixed discriminants of parametrized polynomial systems.
