| Summary: | We study two quite different types of Terracini loci for the order <i>d</i>-Veronese embedding of an <i>n</i>-dimensional projective space: the minimal one and the primitive one (defined in this paper). The main result is that if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>4</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>≥</mo><mn>19</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>≤</mo><mn>2</mn><mi>d</mi></mrow></semantics></math></inline-formula>, no subset with <i>x</i> points is a minimal Terracini set. We give examples that show that the result is sharp. We raise several open questions.
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