Does anybody have a good reference on the Grassmanian and its universal property?


I am reading this paper on Quot schemes: <https://arxiv.org/abs/math/0504590>


Where the Grassmanian is constructed, but its representability and its universal quotient are "exercises". In particular exercise (2) in chapter 1.



Universal property of the Grassmanian
I suggest you:


1. Eisenbud, Harris - *The Geometry of Schemes*, (2000) Springer Verlag, paragraph III.2.7;
2. Eisenbud, Harris - *3264 & All That, Intersection Theory in Algebraic Geometry*, chapters 3 and 4 ([click](http://scholar.harvard.edu/files/joeharris/files/000-final-3264.pdf));
3. Görtz, Wedhorn - *Algebraic Geometry I*, (2010) Vieweg+Teubner Verlag, paragraphs from 8.4 to 8.10;
4. Kleiman S. L. - *Geometry on Grassmannians and Applications to Splitting Bundles and Smoothing Cycles*, Pubblications Mathématiques de l'I.H.É.S., **36** (1969) 281-297 ([clack](http://www.numdam.org/item?id=PMIHES_1969__36__281_0));
5. Vakil - *FOAG* (December 29 2015 version), paragraph 16.7 ([clock](http://math.stanford.edu/~vakil/216blog/FOAGdec2915public.pdf)).


