What's the smallest degree of a homogeneous polynomial that vanishes to order n on a given finite set of points, or more generally on some algebraic variety in projective space? A classical result of Zariski and Nagata tells us the set of such polynomials is the nth symbolic power of the ideal I corresponding to our variety. To bound degrees of elements in the symbolic powers of I, we can look for containments between symbolic powers and other better understood ideals, such as powers of I. We will take a tour through the history of the containment problem and some of its variations, with an eye towards lower bounds for degrees of symbolic powers. Our story will include joint work with Craig Huneke and Vivek Mukundan, and with Sankhaneel Bisui, Tài Huy Hà, and Thái Thành Nguyễn.
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13.1 MB application/pdf
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Symbolic powers, stable containments, and degree bounds
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25031_28419_8367_Symbolic_Powers__Stable_Containments__and_Degree_Bounds.mp4
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