Characterization of projective varieties beyond varieties of minimal degree and del Pezzo varieties

Varieties of minimal degree and del Pezzo varieties are basic objects in projective algebraic geometry. Those varieties have been characterized and classified for a long time in many aspects. Motivated by the question “which varieties are the most basic and simplest except the above two kinds of varieties in view of geometry and syzygies?”, we give an upper bound of the graded Betti numbers in the quadratic strand and characterize the extremal cases. The extremal varieties of dimension $n$, codimension $e$, and degree $d$ are exactly characterized by the following two types: (i) varieties with $d = e+2$, $\operatorname{depth} X =n$, and Green-Lazarsfeld index $a(X)=0$, (ii) arithmetically Cohen-Macaulay varieties with $d = e+3$. This is a generalization of G. Castelnuovo, G. Fano, and E. Park’s results on the number of quadrics and an extension of the characterizations of varieties of minimal degree and del Pezzo varieties in view of linear syzygies of quadrics due to K. Han and S. Kwak. In addition, we show that every variety $X$ that belongs to (i) or (ii) is always contained in a unique rational normal scroll $Y$ as a divisor. Also, we describe the divisor class of $X$ in $Y$.

💡 Research Summary

This paper conducts a systematic study in projective algebraic geometry, aiming to classify the “next simplest” varieties beyond the well-understood classes of varieties of minimal degree (degree d = codimension e + 1) and del Pezzo varieties (ACM with d = e + 2). The authors use the graded Betti numbers in the quadratic strand (β_{p,1}) as a key measure of algebraic complexity.

The primary achievement is the establishment of a new upper bound for β_{p,1} for varieties that are neither of minimal degree nor del Pezzo (Theorem A). This bound is given by β_{p,1}(X) ≤ p * binom(e+1, p+1) - 2 * binom(e, p-1) for 1 ≤ p ≤ e-1. The authors then completely characterize the extremal cases that achieve this new bound (Theorem B). Surprisingly, there are two distinct types: (i) varieties of almost minimal degree (d = e+2) which have depth equal to their dimension n and Green-Lazarsfeld index a(X) = 0 (meaning the first syzygies among quadrics are not all linear), and (ii) arithmetically Cohen-Macaulay (ACM) varieties of degree d = e+3. The proof leverages techniques like inner projection from a general point and the analysis of the syzygies of finite sets of points.

Furthermore, the paper provides a detailed geometric description of these extremal varieties (Theorems C and D). It is shown that any variety belonging to either type (i) or (ii) is always contained as a divisor in a unique (n+1)-dimensional rational normal scroll Y. For type (i), the scroll is of the special form Y = S(0,…,0,1,e-1) and the divisor is linearly equivalent to H + 2F, where H is a hyperplane section and F is a ruling of the scroll. It is also noted that if such a non-conical variety exists, its dimension n is at most 4. For type (ii) ACM varieties of degree e+3, the containing scroll Y = S(a_0,…, a_n) is more general. The divisor class of X in Y is either 2H + (3-e)F (if a_{n-1} > 0) or (e+3)R (if a_{n-1}=0, where R ≅ P^n is an effective generator of the divisor class group). These results translate the algebraic extremality condition into a concrete geometric embedding problem.

In summary, this work significantly extends the classification frontier for projective varieties based on syzygetic properties. It identifies the immediate successors to varieties of minimal and del Pezzo degree in terms of Betti number bounds, reveals an unexpected class (type (i)) sharing these bounds, and situates all extremal varieties within the familiar geometric framework of rational normal scrolls, thereby linking their algebraic simplicity to a specific geometric realization.