PomeTheorem 12.31.3.654B70C2AFD7 For every integer $r\geq 0$,
\[ h^0(M,\operatorname {Sym}^ rT_ M) =1+\frac{d\, r(r+1)(r+2)}6. \]
The tautological line bundle $\mathcal O_{\mathbb P(T_ M)}(1)$ is big and has volume $d$.
Theorem 12.31.3654B70C2AFD7
Proof. By lemma 12.31.1, lemma 12.31.2, the positive weights contribute nothing, the weight zero contributes one dimension, and the weight $-k$ contributes $kd(r-k+1)$ dimensions. Thus
\[ h^0(M,\operatorname {Sym}^ rT_ M) =1+d\sum _{k=1}^ r k(r-k+1) =1+\frac{d\, r(r+1)(r+2)}6. \]
The elliptic curve and its holomorphic line bundle $A$ are algebraic, so $M$ and $\mathbb P(T_ M)$ are projective. The identity $\pi _*\mathcal O_{\mathbb P(T_ M)}(r)=\operatorname {Sym}^ rT_ M$ identifies the section spaces. Since $\mathbb P(T_ M)$ has dimension three,
\[ \operatorname {vol}\bigl (\mathcal O_{\mathbb P(T_ M)}(1)\bigr ) =\limsup _{r\to \infty }\frac{6h^0(M,\operatorname {Sym}^ rT_ M)}{r^3}=d>0, \]
so the tautological line bundle is big. $\square$
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