On boundedness of families of torsion free sheaves

T h eo rem (18 ] Theorem 3. 13 ). L et S be a noe therian schem e and X a projective S -sch em e w ith S am p le in v e rt ib le s h e a f 0 ,( 1 ) . Suppose th at the g e o m e tric f ib re s o f X / S are in te g ral and of dimension n. Then, f o r a f a m i ly g o f the classes o f in v e rtib le sh e av e s on the f ib res o f X / S , the f o llo w in g conditions are equiv alent: (i) g is bounded. (ii) In th e H ilb e rt p o ly n om ia l 2 c ( L ( m ) ) i a , ( m + n — i ) of L e g ., 0 n— i the coe fficient a, is b o u n d e d and a, is bo u n ded b e lo w . (iii) W henev er L runs ov er g , the d e g re e d (c ,(L ) , 0 ,(1 ) ) is b o u n d e d and d (c ,(L ) , C ) ,(1 ) ) is bounded be low .