2CLAUDIURAICU
1. Syzygies of the tangential variety to a rational normal curve
Consider the rational normal curve of degree g, denoted g and defined as the image of the Veronese map 1 g g g 1 g 1 g ⌫ : P P , [a : b] [a : a b : : ab : b ]. (1) g ! ! ··· A first natural question to ask is what are the equations vanishing along g. To answer it, we let S = k[z , ,z ] denote the homogeneous coordinate ring of Pg,letA = k[x, y] denote the homogeneous 0 ··· g coordinate ring of P1, and consider the pull-back homomorphism
g i i : S A, (z )=x y , for i =0, ,g. ! i ···
The ideal I( g) of polynomials vanishing along g is equal to ker( ). Since the matrix xg xg 1y xyg 1 ··· g 1 g 2 2 g "x yx y y # ··· has proportional rows (with ratio x/y), it follows that its 2 2 minors vanish identically. In other words, ⇥ the 2 2 minors of ⇥ z0 z1 zg 1 Z = ··· (2) "z1 z2 zg # ··· belong to ker( )=I( g).
Exercise 1. Show that I( ) is generated by the 2 2 minors of Z. g ⇥ The next natural step is to investigate the minimal free resolution of the homogeneous coordinate ring
S/I( g) of g.Welet S Bi,j( g) = Tori (S/I( g), k)i+j denote the module of i-syzygies of weight j (or degree i + j), and define the Betti numbers of g as
bi,j( g)=dimk Bi,j( g).
The Betti numbers of g are recorded into the Betti table ( g), where the columns account for the homological degree, and the rows for internal degree: i . . j b ( ) ··· i,j g ··· . . KOSZUL MODULES 3
For example, when g = 3, g is the twisted cubic curve, with Betti table 012 0 1–– 1 –32 where a dash indicates the vanishing of the corresponding Betti number. We say that the twisted cubic has a linear minimal free resolution (after the first step), since all (but the first) Betti numbers are concentrated in a single row. This is true more generally for any g, whose minimal free resolution is given by an Eagon–Northcott complex, and the corresponding Betti table takes the following shape: 01 2 i g 1 ··· ··· 0 1– – – – ··· ··· g g g 1 – 2 i (g 1) 2 · 3 ··· · i +1 ··· ✓ ◆ ✓ ◆ ✓ ◆ Remark 2. The description of the defining equations and that of the Betti table of g is independent on the characteristic of the field k.
Let denote the tangential variety of , defined as the union of the tangent lines to . One of the Tg g g problems that we will be concerned with in these notes is the following.
Problem 3. Describe the defining equations and the Betti table of . Tg 1.1. The local description of , . Restricting the Veronese map ⌫ in (1) to the a ne chart x =1 g Tg g yields a local parametrization of g via
t [1 : t : : tg]=(t, t2, ,tg), 7! ··· ··· where [ ] represents projective notation, and ( ) represents a ne notation. The tangent directions ··· ··· to are determined by di↵erentiating with respect to t,so is described in the a ne chart z =1by g Tg 0 2 g g 1 2 g g 1 (t, s) (t, t , ,t )+s (1, 2t, ,gt )=(t + s, t +2ts, ,t + gt s). (3) ! ··· · ··· ··· Since is irreducible, one can use the local description above to check when a homogeneous polynomial Tg belongs to I( ). This observation can be applied to check that the quadratic equations constructed below Tg vanish along . Let Tg zi zj i,j =det , for 0 i