Terracini

Terracini refers primarily to the mathematical concept named after Alessandro Terracini, an Italian mathematician. Specifically, it refers to:

• Terracini's Lemma: A fundamental result in algebraic geometry, particularly concerning the secant varieties of algebraic varieties. Terracini's Lemma provides a way to characterize the tangent space to a secant variety in terms of the tangent spaces to the original variety. More precisely, it states that if a point lies on a secant line (or higher-dimensional secant space) to a variety, then the tangent space to the secant variety at that point is spanned by the tangent spaces to the variety at the points defining the secant line (or space). This lemma is crucial for understanding the geometry of secant varieties and related concepts such as defective varieties. The lemma has different formulations depending on the characteristic of the underlying field, but the general principle remains the same.

Beyond the lemma itself, the name Terracini is sometimes used to refer to related results, generalizations, and applications of Terracini's Lemma in areas such as:

• Secant Variety Computations: The lemma facilitates the computation of the dimension and other properties of secant varieties.
• Defective Varieties: It helps determine whether a given variety is defective (i.e., its secant variety has lower dimension than expected).
• Tensor Decomposition: Connections have been established between Terracini's Lemma and the uniqueness of tensor decompositions.
• Interpolation Problems: Applications appear in the study of polynomial interpolation.

Alessandro Terracini (1889–1968) was an Italian mathematician known for his contributions to differential geometry, algebraic geometry, and analysis. His work on secant varieties and the lemma named after him are foundational results in algebraic geometry.