Effective very ampleness for generalized theta divisors. (English) Zbl 1068.14038

Let \(X\) be a smooth projective complex curve of genus \(g \geq 2\) and fix integers \(r\) and \(d\), with \(r>0\). Let \(\text{U}_X(r,d)\) denote the moduli space of equivalence classes of semistable vector bundles on \(X\) of rank \(r\) and degree \(d\), and let \(\text{SU}_X(r,L)\) denote the moduli space of semistable rank-\(r\) vector bundles of fixed determinant \(L\). Note that the isomorphism class of \(\text{SU}_X(r,L)\) depends only on the degree of \(L\). It is well known that \(\text{Pic}(\text{SU}_X(r,L))\) is isomorphic to \(\mathbb Z\), the ample generator is called the determinant line bundle and denoted by \(\mathcal L\).
Set \(d := \deg(L)\), \(h:= \gcd(r,d)\), \(r_1 := r/h\), \(d_1 := d/h\). For each vector bundle \(F\) on \(X\) of rank \(mr_1\) and degree \(m(r_1(g-1)-d_1)\), there is an associated subscheme of \(\text{SU}_X(r,L)\) parametrizing semistable bundles \(E\) such that \(h^0(E \otimes F) \neq 0\). If the subscheme is not the whole \(\text{SU}_X(r,L)\), then it is the support of a divisor, called a generalized theta divisor and denoted \(\theta _F\). The divisor \(\theta _F\) belongs to \(| \mathcal L ^m |\). If \(m=1\), the divisor \(\theta _F\) is called basic. Also on \(\text{U}_X(r,d)\) divisors \(\theta_F\) can be defined, but they move in the same linear series only if their determinant is kept fixed. The authors prove the following results:
Theorem A : For each \(m \geq r^2 + r\), the linear series \(| \mathcal L ^m |\) on \(\text{SU}_X(r,L)\) separates points and is very ample on the smooth locus \(\text{SU}_X^s(r,L)\). In particular if \(r\) and \(\deg(L)\) are coprime, then \(| \mathcal L ^m |\) is very ample.
Theorem B : Let \(\theta\) be a basic theta divisor on \(\text{U}_X(r,d)\). For each \(m \geq r^2 + r\), the linear series \(| m \theta |\) on \(\text{U}_X(r,d)\) separates points and is very ample on the smooth locus \(\text{U}_X^s(r,d)\). In particular if \(r\) and \(d\) are coprime then \(| m \theta |\) is very ample.
Theorem C : Let \(\theta\) be a basic theta divisor on \(\text{U}_X(r,d)\). If \(d\) is odd, then \(| 3 \theta |\) is very ample. If \(d\) is even and \(X\) is not hyperelliptic, then \(| 5 \theta |\) is very ample.
The authors improve and combine techniques used by E. Esteves [Duke Math. J. 98, 565–593 (1999; Zbl 0983.14028)] and M. Popa [Duke Math. J. 107, 469–495 (2001; Zbl 1064.14032)] and use the dimension estimate for quot schemes obtained by M. Popa and M. Roth [Invent. Math. 152, 625–663 (2003; Zbl 1024.14015)].


14H60 Vector bundles on curves and their moduli
14D20 Algebraic moduli problems, moduli of vector bundles
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