My question is motivated by the following simple observations. By a standard dimensions count in $\mathbb{P}^4$ there should not exist neither an hypersurface of degree $3$ with multiplicity $2$ in seven general points, nor an hypersuperface of degree $5$ with multiplicity $3$ in eight general points. On the other hand:


* through seven general points in $\mathbb{P}^4$ there is a rational normal curve $C$ of degree $4$ and its secant variety $Sec\_2(C)$ is an hypersurface of degree $3$ with multiplicity $2$ along $C$ and in particular in the seven points;
* through eight general points in $\mathbb{P}^4$ there is an elliptic normal curve $E$ and its secant variety $Sec\_2(E)$ is an hypersurface of degree $5$ with multiplicity $3$ along $E$ and in particular in the eight points.


My question is: **does there exist in $\mathbb{P}^4$ an hypersurface of degree $6$ and with multiplicity $4$ in eight general points?** Perhaps the secant variety of some special curve in $\mathbb{P}^4$.



Secant varieties of curves in $\mathbb{P}^4$
The answer is no. You can see this by iterating a standar Cremona transformation. 


Let $p\_1,...,p\_{n+1}\in\mathbb{P}^n$ be general points. We may assume
$$p\_1 = [1:0:...:0],...,p\_{n+1} = [0:...:0:1].$$
We consider the standard Cremona transformation:
$$
\begin{array}{ccc}
\psi:\mathbb{P}^n & \dashrightarrow & \mathbb{P}^n\\
\left[x\_0:...:x\_n\right] & \longmapsto & [\frac{1}{x\_0}:...:\frac{1}{x\_n}]
\end{array}
$$
Note that $\psi\circ \psi = Id\_{\mathbb{P}^n}$, and $\psi^{-1} = \psi$. Let $H\_1,...,H\_{n+1}$ be the coordinate hyperplanes of $\mathbb{P}^n$. Then $\psi$ is not defined on the locus
$$\bigcup\_{1\leq i< j\leq n+1}H\_i\cap H\_j.$$
Furthermore, $\psi$ is an isomorphism off of the union
$$\bigcup\_{1\leq i\leq n+1}H\_i.$$


Let $X\_{n+1}^n$ be the blow-up of $\mathbb{P}^n$ is the $n+1$ base points of the Cremona. Now, $\psi$ induces a birational transformation $\widetilde{\psi}:X\_{n+1}^n\dashrightarrow X\_{n+1}^n$.


Note that, since $\psi$ contracts the hyperplane $H\_i$ passing spanned by the $n$ points $p\_1,...,\hat{p}\_i,...,p\_{n+1}$ to the point $p\_i$, the map $\widetilde{\psi}$ maps the strict transform of $H\_i$ onto the exceptional divisor $E\_i$. Therefore $\widetilde{\psi}$ is an isomorphism in codimension one. Indeed, it is a composition of flops. In particular $\widetilde{\psi}$ induces an isomorphism $Pic(X\_{n+1}^n)\rightarrow Pic(X\_{n+1}^n)$.


Now, the linear system on $\mathbb{P}^n$ associated to the standard Cremona $\psi$ is 
$$\mathcal{H} = \mathcal{O}\_{\mathbb{P}^n}(n)\otimes \mathcal{I}\_{(n-1)(p\_1+...+p\_{n+1})},$$
that is $\mathcal{H}$ is the linear system of hypersurfaces in $\mathbb{P}^n$ of degree $n$ having points of multiplicity at least $n-1$ in $p\_1,...,p\_{n+1}$. Therefore, the inverse image of a general hyperplane of $\mathbb{P}^n$ via $\psi$ is an hypersurface of degree $n$ with points of multiplicity $n-1$ in $p\_1,...,p\_{n+1}$, and
$$\widetilde{\psi}^{\*}H = nH-(n-1)(E\_1+...+E\_{n+1}).$$ 
Furthermore, since $\psi$ contracts the hyperplane $H\_i$ passing spanned by the $n$ points $p\_1,...,\hat{p}\_i,...,p\_{n+1}$ to the point $p\_i$ we have
$$\widetilde{\psi}^{\*}E\_i = H-E\_1-...-\hat{E}\_i-...-E\_{n+1}.$$ 


**Let $D\subset\mathbb{P}^n$ be an hypersurface of degree $d$ having points of multiplicities $m\_1,...,m\_{n+1}$ in $p\_1,...,p\_{n+1}$, and let $\psi:\mathbb{P}^{n}\dashrightarrow\mathbb{P}^n$ be the standard Cremona of $\mathbb{P}^n$. Then 
$$\deg(\psi(D)) = dn-\sum\_{i=1}^{n+1}m\_i$$
and
$$mult\_{p\_i}\psi(D) = d(n-1)-\sum\_{j\neq i}m\_j$$
for any $i = 1,...,n+1$.**


*proof*: Let $X\_{n+1}^n = Bl\_{p\_1,...,p\_{n+1}}\mathbb{P}^n$, and $\widetilde{\psi}:X\_{n+1}^n\dashrightarrow X\_{n+1}^n$ be the birational map induced by $\psi$. The strict transform of $D$ in $X\_{n+1}^n\dashrightarrow X\_{n+1}^n$ can be written as $\widetilde{D} \cong dH-\sum\_{i=1}^{n+1}m\_iE\_i$.\
Now, since $\widetilde{\psi}\_{\*}H = nH-\sum\_{i=1}^{n+1}(n-1)E\_i$, and $\widetilde{\psi}\_{\*}E\_i = H-\sum\_{j\neq i}E\_i$ we get
$$
\begin{array}{ll}
\widetilde{\psi}\_{\*}D = & d(nH-\sum\_{i=1}^{n+1}E\_i)-\sum\_{i=1}^{n+1}m\_i(H-\sum\_{j\neq i}E\_j)=\\
 & dnH-d\sum\_{i=1}^{n+1}(n-1)E\_i-\sum\_{i=1}^{n+1}H+\sum\_{i=1}^{n+1}m\_i\sum\_{j\neq i}E\_j =\\ 
 & (dn-\sum\_{i=1}^{n+1}m\_i)H-\sum\_{i=1}^{n+1}(d(n-1)-\sum\_{j\neq i}m\_j)E\_j.
\end{array} 
$$


*Now, assume there exists an hypersurface $X$ of degree $d = 6$ in $\mathbb{P}^4$ with eight points of multiplicity four. Consider a standard Cremona $f\_1$ centred in five of this eight points. Let $X\_1 = f(X)$. Then $X\_1$ is an hypersurface of degree $d\_1 = 4$ with five points of multiplicity $2$ and three points of multiplicity $4$. Now consider another standard Cremona $f\_2$ centred in three points of multiplicity $2$ of $X\_1$, and two points of multiplicity $4$ of $X\_1$. Then $X\_2 =f\_2(X\_1)$ is an hypersurface of degree $4\cdot 4-4-4-2-2-2 = 2$ with a point of multiplicity $4$. A contradiction.*


