Singularities of secant varieties of rational normal curves

Question

Let $C\subset\mathbb{P}^n$ be a rational normal curve of degree $n$, and let $Sec_k(C)\subset\mathbb{P}^n$ be its $k$-th secant variety. By Theorem 1.1 in this paper:

http://ac.els-cdn.com/S0022404908002387/1-s2.0-S0022404908002387-main.pdf?_tid=120cfede-1405-11e4-91e8-00000aab0f6c&acdnat=1406297453_6fca2d4de380c88c04cfe110390a8418

we have that $Sec_k(C)$ is normal and $Sing(Sec_k(C)) = Sec_{k-1}(C)$. Does $Sec_k(C)$ have ordinary singularities of multiplicity two along $Sec_{k-1}(C)\setminus Sec_{k-2}(C)$?

More precisely let $f:X\rightarrow\mathbb{P}^n$ be the blow-up of $\mathbb{P}^n$ along $Sec_{k-1}(C)$ with exceptional divisor $E$, and let $Y$ be the strict transform of $Sec_k(C)$. Is the following statement true? The strict transform $Y$ is smooth, it intersects $E$ transversally and we have $$Y = f^{*}Sec_k(C)-2E.$$ I guess this should be true for instance when we consider a rational normal curve $C$ of degree four and $Sec_2(C)$ which is a cubic hypersurface.

Answer 1

The following is a consequence of Theorem 1 in "A. Bertram, Moduli of Rank-$2$ Vector Bundles, Theta divisors, and the geometry of curves in projective space, J. Differential Geom. 35, 1992, 429-469."

Let $C\subset\mathbb{P}^{2h}$ be a degree $2h$ rational normal curve. Consider the following sequence of blow-ups:

• $\pi_1:X_1\rightarrow\mathbb{P}^{2h}$ the blow-up of $C$,

• $\pi_2:X_2\rightarrow X_{1}$ the blow-up of the strict transform of $Sec_2(C)$,

  $\vdots$

• $\pi_{h-1}:X_{h-1}\rightarrow X_{h-2}$ the blow-up of the strict transform of $Sec_{h-1}(C)$.

Let $\pi:X\rightarrow\mathbb{P}^{2h}$ be the composition of these blow-ups. Then, for any $k\leq h$ the strict transform of $Sec_{k-1}(C)$ is smooth, irreducible and transverse to all exceptional divisors. In particular $Y$ is smooth and the divisor in $Y$ given by the union of the exceptional divisors and the strict transform of $Sec_{h}(C)$ is simple normal crossing.

It is enough to apply Theorem 1 of the above cited paper and observe that The rational normal curve is given by the Veronese embedding induced by the line bundle $L = \mathcal{O}_{\mathbb{P}^1}(2h)$ on $\mathbb{P}^1$. Now, $$h^{0}(\mathbb{P}^1,L(-2h)) = 1 = 2h+1-2h= h^{0}(\mathbb{P}^1,L)-2h.$$ This means that $C\subset\mathbb{P}^{2h}$ is embedded by a $2h$-very ample line bundle.