For every \(e \geqslant-1\) and \(n \geqslant e+3\), there is an elliptic scroll of degree \(d=2 n-e\) in \(\PP^{d-1}\). In particular, there is an elliptic scroll of degree 5 in \(\PP^4\).
Solution
Fix an elliptic curve \(C\).
1. For every integer\[ e\ge-1 \]there is a ruled surface\[ \pi:X=\PP(\mathcal E)\longrightarrow C \]with invariant \(e\).
For \(e=-1\), Hartshorne V.2.5(c), proved on P-AGH525INVARIANTE, gives a ruled surface over the genus-one curve \(C\) with invariant \(-1\).
For \(e=0\), V.2.5(a) gives an indecomposable normalized rank-two bundle with invariant zero.
Now let \(e>0\). Choose a line bundle \(M\) of degree \(-e\) on \(C\) and put \[ \mathcal E=\OO_C\oplus M. \] The summand \(\OO_C\) gives a nonzero global section. If \(N\) has negative degree, then both \[ N \qquad\text{and}\qquad M\tensor N \] have negative degree and hence no sections. Thus \[ H^0(C,\mathcal E\tensor N)=0, \] so \(\mathcal E\) is normalized. Since \[ \deg\det\mathcal E=-e, \] the invariant of \(\PP(\mathcal E)\) is \(e\).
2. Let \(C_0\) be the normalized section and \(f\) a ruling fibre. Then\[ C_0^2=-e, \qquad C_0\cdot f=1, \qquad f^2=0. \]
These are the standard intersection formulas for a normalized geometrically ruled surface: the invariant is \[ e=-C_0^2, \] while a section meets every fibre once and distinct fibres are disjoint.
3. Choose a divisor \(\mfb\) on \(C\) with\[ \deg\mfb=n, \]and put\[ D=C_0+\mfb f. \]If\[ n\ge e+3, \]then \(D\) is very ample.
This is exactly Hartshorne V.2.12(b), proved on P-AGH5212ELLRULEDAMPLE: on a ruled surface over an elliptic curve, \[ |C_0+\mfb f| \] is very ample if and only if \[ \deg\mfb\ge e+3. \]
4. The embedding defined by \(|D|\) makes every ruling fibre a line.
By step 2, \[ D\cdot f = (C_0+\mfb f)\cdot f = 1. \] Hence \[ \OO_X(D)|_f\cong\OO_{\PP^1}(1). \] Because the complete system \(|D|\) is very ample by step 3, its restriction embeds each fibre as a projective line. Thus the image is a scroll over the elliptic curve \(C\).
5. The degree of this embedded scroll is\[ \boxed{d=2n-e}. \]
For a surface embedded by the very ample divisor \(D\), its degree is \(D^2\). Using step 2, \[ \begin{aligned} D^2 &=(C_0+\mfb f)^2\\ &=C_0^2+2n(C_0\cdot f)+n^2f^2\\ &=-e+2n\\ &=2n-e. \end{aligned} \]
6. The complete linear system \(|D|\) has\[ \boxed{h^0(X,\OO_X(D))=2n-e=d}. \]
Let \(\mfe\) be the divisor on \(C\) with \[ \OO_C(\mfe)\cong\det\mathcal E. \] Then \[ \deg\mfe=-e. \]
The section restriction calculation from Hartshorne V.2.11, proved in P-AGH5211VERYAMPLESECT, gives, because \(\mfb\) is nonspecial, \[ h^0(X,\OO_X(C_0+\mfb f)) = h^0(C,\OO_C(\mfb)) + h^0(C,\OO_C(\mfb+\mfe)). \] Here \[ \deg\mfb=n>0, \qquad \deg(\mfb+\mfe)=n-e\ge3>0. \] On the elliptic curve, Riemann–Roch and Serre duality give \[ h^0(C,L)=\deg L \] for every positive-degree line bundle \(L\). Therefore \[ h^0(X,\OO_X(D)) = n+(n-e) = 2n-e = d. \]
7. The complete linear system \(|D|\) embeds \(X\) as an elliptic scroll ofdegree \(d\) in\[ \boxed{\PP^{d-1}}. \]
Step 3 gives very ampleness, so the complete system defines a closed immersion \[ X\hookrightarrow\PP\bigl(H^0(X,\OO_X(D))^\vee\bigr). \] By step 6 the target is \(\PP^{d-1}\). Step 4 identifies the image as a scroll over the elliptic curve, and step 5 gives degree \(d\).
8. There is an elliptic scroll of degree \(5\) in \(\PP^4\).
Take \[ e=-1, \qquad n=2. \] Then \[ n=e+3, \] so the construction applies, and \[ d=2n-e=4+1=5. \] Step 7 therefore gives an elliptic scroll of degree \(5\) in \[ \PP^{5-1}=\PP^4. \]
9. Q.E.D.
Steps <1>1–<1>7 prove the asserted existence for every \(e\ge-1\) and \(n\ge e+3\), and step 8 gives the stated degree-five example.