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

Esteves, Eduardo; Popa, Mihnea

Duke Math. J. 123, No. 3, 429-444 (2004).

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)].

Reviewer: Luciana Ramella (Genova)

MSC:

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

Keywords:

semistable vector bundles; moduli spaces; generalized theta divisors; Quot schemes

Citations:

Zbl 0983.14028; Zbl 1024.14015; Zbl 1064.14032

References:

[1]	L. Brambila-Paz, I. Grzegorczyk, and P. E. Newstead, Geography of Brill-Noether loci for small slopes , J. Algebraic Geom. 6 (1997), 645–669. · Zbl 0937.14018
[2]	S. Brivio and A. Verra, The theta divisor of \(\SU_C(2,2d)^s\) is very ample if \(C\) is not hyperelliptic , Duke Math. J. 82 (1996), 503–552. · Zbl 0876.14024
[3]	–. –. –. –., On the theta divisor of \(\SU(2,1)\) , Internat. J. Math. 10 (1999), 925–942. · Zbl 1077.14536
[4]	R. Donagi and L. W. Tu, Theta functions for \(\text \mathrm SL(n)\) versus \(\text \mathrm GL(n)\) , Math. Res. Lett. 1 (1994), 345–357. · Zbl 0847.14027
[5]	J.-M. Drezet and M. S. Narasimhan, Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques , Invent. Math. 97 (1989), 53–94. · Zbl 0689.14012
[6]	E. Esteves, Separation properties of theta functions , Duke Math. J. 98 (1999), 565–593. · Zbl 0983.14028
[7]	G. Faltings, Stable \(G\)-bundles and projective connections , J. Algebraic Geom. 2 (1993), 507–568. · Zbl 0790.14019
[8]	B. van Geemen and E. Izadi, The tangent space to the moduli space of vector bundles on a curve and the singular locus of the theta divisor of the Jacobian , J. Algebraic Geom. 10 (2001), 133–177. · Zbl 0989.14010
[9]	S. L. Kleiman, The transversality of a general translate , Compositio Math. 28 (1974), 287–297. · Zbl 0288.14014
[10]	H. Lange, Zur Klassifikation von Regelmannigfaltigkeiten , Math. Ann. 262 (1983), 447–459. · Zbl 0492.14003
[11]	G. Pareschi and M. Popa, Regularity on abelian varieties, I , J. Amer. Math. Soc. 16 (2003), 285–302. · Zbl 1022.14012
[12]	M. Popa, Dimension estimates for Hilbert schemes and effective base point freeness on moduli spaces of vector bundles on curves , Duke Math J. 107 (2001), 469–495. · Zbl 1064.14032
[13]	–. –. –. –., Verlinde bundles and generalized theta linear series , Trans. Amer. Math. Soc. 354 (2002), 1869–1898. JSTOR: · Zbl 0996.14015
[14]	M. Popa and M. Roth, Stable maps and Quot schemes , Invent. Math. 152 (2003), 625–663. · Zbl 1024.14015
[15]	M. Teixidor i Bigas and L. W. Tu, “Theta divisors for vector bundles” in Curves, Jacobians, and Abelian Varieties (Amherst, Mass., 1990) , Contemp. Math. 136 , Amer. Math. Soc., Providence, 1992, 327–342. · Zbl 0783.14020