On the Singular Loci of Higher Secant Varieties of Veronese Embeddings

Abstract The 𝑘-th secant variety of a projective variety <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>X</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mi>N</m:mi> </m:msup> </m:mrow> </m:math> X\subset\mathbb{P}^{N} , denoted by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>σ</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>X</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \sigma_{k}(X) , is defined to be the closure of the union of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>k</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (k-1) -planes spanned by 𝑘 points on 𝑋. In this paper, we examine the 𝑘-th secant variety <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi>σ</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mi>v</m:mi> <m:mi>d</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>⊂</m:mo> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mi>N</m:mi> </m:msup> </m:mrow> </m:math> \sigma_{k}(v_{d}(\mathbb{P}^{n}))\subset\mathbb{P}^{N} of the image of the 𝑑-uple Veronese embedding <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>v</m:mi> <m:mi>d</m:mi> </m:msub> </m:math> v_{d} of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> \mathbb{P}^{n} to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mi>N</m:mi> </m:msup> </m:math> \mathbb{P}^{N} with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>N</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mfrac linethickness="0pt"> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mi>d</m:mi> </m:mrow> <m:mi>d</m:mi> </m:mfrac> <m:mo>)</m:mo> </m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mrow> </m:math> N=\binom{n+d}{d}-1 , and focus on the singular locus of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>σ</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mi>v</m:mi> <m:mi>d</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:m