**Answer under revision.** The OP has now asked for the proof of irreducibility for all $e\geq 1$, not just for $e= 2.$ The argument for that uses the same basic ideas, but the edits are getting very long. As I have time, I will write this up as a PDF file and add a link to this post. What is written below is provisional, until I can finish the PDF document.

The arguments below are roughly the same as in the second chapter of my thesis. My thesis included some *ad hoc* arguments that apply for cubic threefolds. However, I think it is better to base the argument on a slight variant of Cristian Minoccheri's version of the Bertini Connectedness Theorem, Proposition 3.1 of the following.

Cristian Minoccheri

*On the Arithmetic of Weighted Complete Intersections of Low Degree*

https://arxiv.org/pdf/1608.01703.pdf

The following variant combines Minoccheri's result with Lemma 3.1 of the following.

A. J. de Jong and Jason Starr

*Low degree complete intersections are rationally simply connected*

https://www.math.stonybrook.edu/~jstarr/papers/nk1006g.pdf

**Variant Statement.** Let $X$ be a smooth $k$-scheme that is integral and étale simply connected. Let $M$ and $N$ be normal integral schemes. Let $i:N\to M$ and $h:M\to X$ be proper, finitely presented morphisms such that both $h$ and $h\circ i$ are surjective. For the smooth locus $M^o$ of $h$, if the complement of $i^{-1}(M^o)$ has codimension $\geq 2$ in $N$, then all fibers of $h$ are connected.

**Proof.** This is essentially the same proof as in Minoccheri's theorem. For the finite parts, $u:\widetilde{M}\to X$, resp. $v:\widetilde{N}\to X$, of $h$, resp. of $h\circ i$, there is an induced $X$-morphism $\widetilde{i}:\widetilde{N}\to \widetilde{M}$. For the smooth locus $\widetilde{M}^o$ of $u$, the inverse image in $N$ of $\widetilde{M}^o$ contains the inverse image of $M^o$. By hypothesis, the complement has codimension $\geq 2$. Thus, by the usual Seidenberg argument (e.g., Section 15 of Matsumura's "Commutative ring theory"), also the complement of $\widetilde{i}^{-1}(\widetilde{M}^o)$ in $\widetilde{N}$ has codimension $\geq 2$. Since $\widetilde{i}$ is a finite, surjective morphism of normal, integral schemes, the Going-Up and Going-Down Theorem hold. Therefore, the singular locus of $u$ has codimension $\geq 2$. By the Purity Theorem, Section X.3 of SGA 2, $u$ is finite and étale. By hypothesis, $X$ is étale simply connected. Thus $u$ is an isomorphism. So every (geometric) fiber of $h$ is connected. **QED**

**Definition.** For a closed subscheme $Y$ of $\mathbb{P}^n$, the *nonlinear locus* is the maximal open substack $\mathcal{M}_{0,r}(Y,e)^o\subset \mathcal{M}_{0,r}(Y,e)$ parameterizing stable maps with irreducible domain whose image is contained in no line.

**Application 1.** Irreducibility of the geometric generic fiber of $\text{ev}:\mathcal{M}_{0,1}(Y,2)^o \to Y$ for $Y\subset \mathbb{P}^n$ a general hypersurface of degree $n-1$.

**Proposition [Harris, Roth, S].** For $Y\subset \mathbb{P}^n$ a sufficiently general hypersurface of degree $d\leq n-1$, $\mathcal{M}_{0,1}(Y,1)$ is irreducible and smooth of the expected dimension $2n-d-2$ and the evaluation morphism $\text{ev}:\mathcal{M}_{0,1}(Y,2)\to Y$ is flat of relative dimension $n-d-1$.

**Sketch of proof.** Inside the parameter space of Taylor expansions of a degree $d$ polynomial about a point of $\mathbb{P}^n$ that is a zero of the polynomial, for the closed locus where the common zero locus of the $d$ homogeneous parts has codimension strictly less than $d$, the codimension of this locus is $\geq n$ by a specialization argument. Since $Y$ has dimension $n-1$, this locus is disjoint from $Y$ for a general choice of $Y$. **QED.**

Let $n\geq 4$ be an integer. Let $d$ equal $n-1$.

**Notation.** Denote by $\mathcal{Y}$, resp. $\mathcal{Y}'$, the projective space parameterizing degree $d$ hypersurfaces $Y$ in $\mathbb{P}^n$, resp. pairs $(q,[Y])$ of $[Y]\in \mathcal{Y}$ and a point $q\in Y$. Denote by $\mathcal{W}$, resp. $\mathcal{W}'$, the parameter space of pairs $([C],[P])$ of a $2$-plane $P$ in $\mathbb{P}^n$ and a plane conic $C$ in $P$, resp. of triples $(q,[C],[P]).$ Denote by $\mathcal{V}\subset \mathcal{W}\times \mathcal{Y}$, resp. by $\mathcal{V}'\subset \mathcal{W}'\times \mathcal{Y}$, the parameter space of triples $(([C],[P]),[Y])$ such that $C$ is contained in $Y$, resp. of $4$-tuples $((q,[C],[P]),[Y])$ such that $C$ is contained in $Y$. Denote by $\Theta:\mathcal{V}\to \mathcal{Y}$, resp. $\Theta':\mathcal{V}'\to \mathcal{Y}',$ the natural projection.

**Lemma.** The closed subscheme $\mathcal{V}$ with its projection to $\mathcal{W}$ is a projective subbundle of codimension $2d+1$ of the product projective bundle $\mathcal{W}\times \mathcal{Y}$, and similarly for the base change $\mathcal{V}'\to \mathcal{W}'$. In particular, $\mathcal{V}$, resp. $\mathcal{V}'$, is irreducible and it is smooth over $\mathcal{W}$, resp. over $\mathcal{W}'.$

**Proof.** This is equivalent to the statement that every $C$ parameterized by $\mathcal{W}$ "imposes independent conditions" on degree $d$ polynomials, i.e., the following restriction homomorphism is surjective, $$ H^0(\mathbb{P}^n,\mathcal{O}(d))\to H^0(C,\mathcal{O}(d)|_C).$$ This is equivalent to the statement that for the ideal sheaf $\mathcal{I}$ of $C$ in $\mathbb{P}^n$, the cohomology $h^1(\mathbb{P}^n,\mathcal{I}(d))$ is zero. Since $C$ is a complete intersection curve of type $(1,\dots,1,2)$, by the Koszul resolution, $h^q(\mathbb{P}^n,\mathcal{I}(e))$ equals zero for all $q>0$ and all $e\geq 0$. **QED**

**Notation.** For every locally closed subscheme $D$ of $\mathcal{W}$, resp. of $\mathcal{W}'$, denote by $\mathcal{V}_D$ the inverse image of $D$ in $\mathcal{V}$, resp. denote by $\mathcal{V}'_D$ the inverse image of $D$ in $\mathcal{V}'.$

**Lemma.** For every irreducible, locally closed subscheme $D$ of $\mathcal{W}$, resp. of $\mathcal{W}'$, that is smooth or even just regularly embedded, for the restriction $\Theta_D:\mathcal{V}_D\to \mathcal{Y},$ resp. $\Theta'_D:\mathcal{V}'_D\to \mathcal{Y}'$, the morphism $\Theta,$ resp. $\Theta',$ is smooth at every smooth point of $\Theta_D,$ resp. at every smooth point of $\Theta'_D.$

**Proof.** Since $\mathcal{W}$, resp. $\mathcal{W}'$ is smooth, the locally closed subscheme is a regular immersion of some codimension $c$. Since $\mathcal{V}\to \mathcal{W}$, resp. $\mathcal{V}'\to \mathcal{W}',$ is flat (even smooth), also $\mathcal{V}_D$ is a regular immersion of codimension $c$ in $\mathcal{V}$, resp. $\mathcal{V}'_D$ is a regular immersion of codimension $c$ in $\mathcal{V}'.$ **QED**

**Lemma.** For the locus $D$ in $\mathcal{W}$ parameterizing nonreduced conics ("double lines"), the morphism $\mathcal{V}_D\to \mathcal{Y}$ is generically smooth. Thus $\Theta$ is smooth at the generic point of $\mathcal{V}_D$.

**Proof.** One proof of this uses that for every $b$ with $2<b\leq n$, every smooth Fano hypersurface $Y$ of degree $b$ in $\mathbb{P}^n$ contains lines $L$ with $h^1(N_{L/Y}(-1)) >0.$ This follows by a parameter count (also Pandharipande has a proof of this using enumerative techniques). Thus $N_{L/Y}$ has a subbundle $\mathcal{O}(1)$. This subbundle of $N_{L/\mathbb{P}^n}$ equals $N_{L/P}$ for a unique $2$-plane $P$ that contains $L$ and is contained in $\mathbb{P}^n$. The intersection of $P$ with $Y$ contains a conic $C$ supported on $L$. Therefore $\mathcal{V}_D\to \mathcal{Y}$ is surjective. Now apply the previous lemma. **QED**

**Lemma 2.** For the universal family of pointed lines over $\mathcal{Y}'$, $\text{ev}_1:\overline{\mathcal{M}}_{0,1}(\mathcal{Y}'/\mathcal{Y},1) \to \mathcal{Y}'$, the unique irreducible component of the singular locus of $\ev_1$ that has codimension $1$ is irreducible and a general point of the singular locus parameterizes a triple with $Y$ smooth along $L$ and with $N_{L/Y}$ containing a unique subbundle $\mathcal{O}(1)$. The morphism $\text{ev}_1$ is generically finite.

**Proof.** First, it is $n-1$ conditions on $Y$ to be singular at some point of $L$. Since $n-1>1$, it suffices to consider triples such that $Y$ is smooth along $L$. Since $N_{L/\mathbb{P}^n}$ is $\mathcal{O}(1)^{\oplus(n-1)}$, and since $N_{Y/\mathbb{P}^n}|_L \cong \mathcal{O}(n-1),$ the bundle $N_{L/\mathbb{P}^n}(-1)$ has nonzero $h^1$ if and only if there is a subbundle isomorphic to $\mathcal{O}(1)$. This subbundle is $N_{L/P}$ for a unique $2$-plane $P$ in $\mathbb{P}^n$ that contains $L$. The parameter space of such $2$-plane is $\mathbb{P}^{n-2}$. For fixed $L$ and $P$, it is $d$ linear conditions on $Y$ for $N_{L/P}$ to be a subbundle of $N_{L/Y}$. Thus the singular locus is the image of a projective bundle over a $\mathbb{P}^{n-2}$-bundle over the flag variety $\text{Flag}(1,2;n+1)$. Counting parameters, this iterated projective bundle has dimension $1$ less than the dimension of $\overline{\mathcal{M}}_{0,1}(\mathcal{Y}'/\mathcal{Y},1).$ For fixed $(p,[L],[Y])$, if there are two or more choices of $P$, then there is a positive dimensional family of such $P$. Thus, the image of this locus in $\overline{\mathcal{M}}_{0,1}(\mathcal{Y}'/\mathcal{Y},1)$ has codimension $\geq 2$. **QED.**

Inside $\mathcal{X}\setminus \mathcal{X}_D$, denote by $\Delta$ the closed subset where $C$ is a union of two lines $L\cup L'$ with $(q,[L],[Y])$ and $(q,[L'],[Y])$ being singular points of $\Psi$.

**Lemma 3.** The codimension $1$ part of $\Delta$ is irreducible. The intersection of $\mathcal{X}\setminus \mathcal{X}_D$ with $\text{Sing}(\Phi)$ is a proper closed subset of this irreducible, codimension $1$ subset of $\mathcal{X}\setminus \mathcal{X}_D$.

**Proof.** The open subset $\mathcal{Z}\setminus \mathcal{Z}_D$ is a single orbit under $\text{Aut}(\mathbb{P}^n)$ parameterizing conics that are unions $L\cup L'$ of a pair of distinct lines in $P$. The locus where $Y$ is singular at a point of $L$, resp. $L'$, has codimension $n-1$. Since $n\geq 4$, this is $\geq 2$. Thus, it suffices to consider the case where $Y$ is smooth along $C$. By Lemma 2, it is a codimension $2$ condition for $N_{L/Y}$ to have worse than a single $\mathcal{O}(1)$, and similarly for $L'$. So it suffices to consider the locus where both $N_{L/Y}$ and $N_{L'/Y}$ have a subbundle $\mathcal{O}(1)$ and no worse.

Denote by $P$, resp. $P'$, the unique $2$-plane that contains $L$, resp. $L'$, with $N_{L/P}$ contained in $N_{L/Y}$, resp. with $N_{L'/P'}$ contained in $N_{L'/Y}$. There is a $\mathbb{P}^{d-1}$ parameterizing $P$, and there is a $\mathbb{P}^{d-1}$ parameterizing $P'$. Except for the degenerate case when $P$ equals $P'$, which has higher codimension, for fixed $P$ and $P'$, it is $2d-1$ linear conditions on $Y$ for $N_{L/P}$ to be contained in $N_{L/Y}$ and for $N_{L'/P'}$ to be contained in $N_{L'/Y}$. The parameter space for such data is the generically finite image of the total space of a projective bundle over the $\mathbb{P}^{d-1}\times \mathbb{P}^{d-1}$-bundle over $\mathcal{Z}\setminus \mathcal{Z}_D$. This image is irreducible.

Finally, for a pair as above, since $N_{C/Y}|_L$ is the elementary transform up of $N_{L/Y}$ at the point $q$ in the normal direction $T_q L'$ of $L'$, $N_{C/Y}(-1)$ has vanishing $h^1$ unless $T_q L'$ is in the subbundle of $N_{L/Y}$ generated by global sections, cf. Graber-Harris-Starr. This is one additional linear condition on $L'$. Thus, the locus is a proper closed subset of $\Delta$. **QED**

**Theorem.** The morphism $\Phi:\mathcal{X}\to \mathcal{Y}$ is a generically smooth morphism between smooth, projective varieties. The target $\mathcal{Y}$ is a projective space, thus algebraically simply connected. The singular locus of $\Phi$ has codimension $\geq 2$ everywhere in $\mathcal{X}$. The geometric generic fiber of $\Phi$ is irreducible.

**Proof.** The intersection of the singular locus with $\mathcal{X}_D$ has codimension $\geq 2$ in $\mathcal{X}$ by Lemma 1. The intersection of the singular locus with $\mathcal{X}\setminus \mathcal{X}_D$ is a proper closed subset of $\Delta$. The codimension $1$ part of $\Delta$ is irreducible. Thus, $\text{Sing}(\Phi)$ has codimension $2$ in $\mathcal{X}\setminus \mathcal{X}_D$. By Cristian Minoccheri's version of the Bertini Connectedness Theorem, the geometric generic fiber of $\Phi$ is connected. It is also LCI, since $\Phi$ is a surjective morphism between smooth schemes. Finally, by the computation above, the singular locus of the fiber has codimension $\geq 2$. Thus, the geometric generic fiber of $\Phi$ is normal by Serre's criterion. Since it is also connected, it is irreducible. **QED**

**Original post.**I cannot promise that the following is the argument that we had at that time, but this is how we used to prove similar results. Let $Y$ be a general hypersurface of degree $d$ in $\mathbb{P}^n$. Consider the evaluation morphism from the parameter space of pointed lines to $Y$, $$f: X \to Y.$$ This is a finite, surjective morphism from a smooth variety $X$ to a smooth variety $Y$ that is branched over a divisor $B$. By parameter counts (similar to the ones in Chapter V of Kollár's book), the morphism has simple branching at general points of $B$ if $Y$ is general, i.e., the locus where $h^1$ of the normal bundle is $\geq 2$ has codimension $2$ in the parameter space $X$ of pointed lines in $Y$, and the locus of line pairs both having positive $h^1$ of the normal bundle is also codimension $2$ in the parameter space $\text{Sym}^2(X/Y)$ of line pairs in $Y$: for a fixed line, resp. line pair, in $\mathbb{P}^n$, consider the linear conditions on the partial derivatives of a defining equation of $Y$ in order for $h^1$ to be $\geq 2$, resp. for $h^1$ to be $\geq 1$ for both lines.

Thus, by the same answer as in the following MathOverflow question, the restriction of this finite morphism over a general $2$-plane section of $Y$ has full monodromy.

Monodromy representation of elementary simple covers

To say one more word about this: the monodromy group $H$ of the cover is a transitive subgroup of $\mathfrak{S}_n$ that is generated by transpositions. So all elements of $\{1,\dots,n\}$ are equivalent, the partition is trivial, and $H$ is the full symmetric group.

Since the parameter space of line pairs is the (closure of the) complement of the diagonal in the relative $\text{Sym}^2$ of the morphism $f$ with its projection to $Y$, this monodromy result implies that the inverse image of a general $2$-plane section of $Y$ in the space of line pairs is irreducible. Thus, the parameter space of line pairs is irreducible.

**Edit to original answer.** The OP asks the valid question: how do we know that the branching is simple over $B$? Consider the incidence correspondence $$\mathcal{X} = \{ (p,[L],[Y]) : p\in L\subset Y\},$$ where $L$ is a line and $Y$ is a hypersurface of degree $d$ in $\mathbb{P}^n$, $n=d+1$. For the incidence correspondence $\mathcal{Y}=\{(p,[Y]) : p\in Y\}$, there is a universal forgetful morphism, $$\Phi:\mathcal{X}\to \mathcal{Y}, \ \ (p,[L],[Y])\mapsto (p,[Y]).$$ There is an open dense subset $\mathcal{X}^0$ of $\mathcal{X}$ parameterizing triples such that $Y$ is smooth along $L$. Via projection to the flag variety of pointed lines, the locus in $\mathcal{X}^0$ where $N_{L/Y}$ has nonzero $h^1$ is an irreducible divisor $D$ in $\mathcal{X}^0$. The open subset $D^0$ where $h^1$ equals $1$ is open in $D$. It suffices to find an arc in $\mathcal{X}$ that intersects $D^0$ in one point and such that the branching of $\Phi$ along that arc is simple.

Choose homogeneous coordinates on $\mathbb{P}^n,$ $[x,y,z,w,u_4,\dots,u_n]$. Inside the projective $3$-space, $\mathbb{P}^3=\text{Zero}(u_4,\dots,u_n)$, consider the following singular cubic surface, $$\Sigma = \text{Zero}(H), \ \ H = xz^2-yw^2.$$ For every $[s,t]\in \mathbb{P}^1$, consider the following line in $\Sigma,$ $$L_{s,t} = \text{Zero}(tw-sz,t^2x-s^2y).$$ Consider this as a pointed line via the point $p_{s,t}=[s^2,t^2,0,0].$ The union of these lines is a line $M=\text{Zero}(z,w)$. For each $p\in M$ other than $[1,0,0,0]$ and $[0,1,0,0]$, there are precisely two values $[s,t]$ and $[s,-t]$ such that the corresponding point equals $p$. Thus, as $p$ in $M$ approaches $[1,0]$, this is a family of pointed lines with simple branching.

Now consider the following degree $d$ polynomial, $$F(x,y,z,w,u_4,\dots,u_{d+1}) = x^{d-3}H + u_4(x^{d-1}+w^{d-1}) + $$ $$\left[u_5x^{d-4}w^3 + \dots + u_{d+1}w^{d-1}\right].$$ The last terms in brackets are only included if $d\geq 4$. It is straightforward to compute that $Y=\text{Zero}(F)$ is smooth along the line $L_{1,0}$, and the normal bundle of the line in $Y$ is $\mathcal{O}(-1)\oplus \mathcal{O}^{\oplus(d-3)}\oplus \mathcal{O}(1)$. Thus the family of pointed lines gives an arc in $D^0$. So the fiber of $f$ is curvilinear at this point. We have the two branches $L_{s,t}$ and $L_{s,-t}$ permuted by the local monodromy. If there were any worse branching on the branch containing $(p_{1,0},[L_{1,0}])$, then the fiber would not be curvilinear.

To avoid any mention of branching (this is somewhat problematic in positive characteristic), it should be possible to compute the Hessian of $\Phi$ at the point $(p_{1,0},[L_{1,0}],Y)$ and see that this point is a simple ordinary double point of the fiber of $\Phi$.