Talk 1: Syszigies and singularities of secant varieties
Definition
Let \(X\subset \mathbb P^r\) be smooth. The secant variety of \(X\) is \[\Sigma(X) = \overline{\bigcup_{p_1\neq p_2} \text{span}\{p_1,p_2\}}\]
1) Singularities.
It’s a theorem of Terracini that for \(p\in \text{span}\{p_1,.p_2\}\), \(\text{span}\{T_{p_1}X, T_{p_2}X\} \subseteq T_p\Sigma(X)\). Moreover if \(p,p_1\) and \(p_2\) are general then equality holds.
Speaker asks the question: If \(X\subseteq \text{Sing}~\Sigma(X)\),
- when does equality in Terracini’s theorem hold?
- Is \(\Sigma(X)\) normal?
- If \(\Gamma\)is a \(\mathbb Q\)-divisor, is \((\Sigma(X), \Gamma)\) lc?
2) Geometry of subvarieties in projective space.
Conjecture
(Hartshorne) If \(3\dim(X) > 2r\), then \(X\) is a complete intersection (\(\dim(X) > 2\codim(X)\)).
Definition
\(X\subseteq \mathbb P^r\) is projective normal if for all \(m > 0\) we have \[H^0(\mathbb P^r, \mathcal O_{\mathbb P^r}(m)) \twoheadrightarrow H^0(X, \mathcal O_X(m)).\]
Conjecture
\(X\) is projective normal if \(3\dim(X) > 2r\).
Theorem
(Zak). If \(3\dim(X) > 2(r - 2)\) then \(\Sigma(X) = \mathbb P^r\).
Corollary
If \(3\dim(X) > 2(r - 1)\) then \(X\) is linearly normal, i.e. \[H^0(\mathbb P^r, \mathcal O_{\mathbb P^r}(1)) \twoheadrightarrow H^0(X, \mathcal O_X(1)).\]
Proof
Suppose not. Then \(X\) embeds in \(\mathbb P^r\) and \(\mathbb P^{r+1}\) in such a way that fits in a commutative diagram with a rational map \(\mathbb P^{r+1} \to \mathbb P^r\).
3) Moduli of vector bundles.
Definition
\(\Sigma_k(X) = \overline{\bigcup_{p_1,...,p_k}\text{span}\{p_1,...,p_k\}}\).
If \(C\) is a smooth curve, \(\mathcal L\) a line bundle of degree \(d\) on \(C\), then \[\mathcal M_{2,\mathcal L} = \{\text{rank 2 semistable vector bundles } E, \det E \simeq \mathcal L.\}\]
He then writes
\begin{aligned} \{0\to \mathcal O_X\to E\to \mathcal L \to 0\} &\leftrightarrow \Ext^1(\mathcal L, \mathcal O_C) \\ &\cong H^1(C, \mathcal L^\vee) \\ &\cong H^0(C, \omega_C\otimes \mathcal L)^\vee. \end{aligned}Write \[\phi_L:\mathbb PH^0(\omega_C\otimes L)^\vee \to \mathcal M_{2, L}\]
Missed stuff, ends section with a theorem of Bertram (cool):
Theorem
(Bertram) When \(M\) is a line bundle on \(C\) with \(\deg M \geq 2g + 1 + 2k\), then \(\text{Sing}\Sigma_k(C) = \Sigma_{k-1}(C)\).
4) Complexity theory.
Question: How computationally involved is it to multiply two \(n\times n\) matrices?
The multiplication map \(m:M_{n\times n}\times M_{n\times n} \to M_{n\times n} \) is a tensor in \(M_{n\times n}\vee \otimes M_{n\times n}^\vee \otimes M_{n\times n} \).
If we set \(S = \mathbb P M_{n\times n}\times \mathbb P M_{n\times n}\times \mathbb P M_{n\times n} \hookrightarrow \mathbb P(M_{n\times n}\otimes M_{n\times n}\otimes M_{n\times n})\), then the answer is that it is proportional to \(\min\{k ~ | ~ m \in \Sigma_k(S)\}\).
Talk 2: Plus Pure Thresholds of some cusp-like singularities (Kevin Tucker)
This is a mixed characteristic analog of the log-canonical threshold, apparently a gold standard invariant in this setting. First we start with that.
Start in characteristic 0, take \(f\in \mathbb C[x_1,...,x_n]\), \(f\neq 0\). Let \[\text{lct}(f) := \sup\left\{t\in \mathbb R_{>0} ~ | ~ \int_{\text{near } 0} \frac{1}{|f|^{zt}} < \infty\right\}\]
e.g. \(\text{lct}(x^m) = \frac1m\), easy by using polar coordinates, \(\text{lct}(x_1^{m_1}\cdot ...\cdot x_n^{m_n}) = \min\{1/m_i\}\).
The idea is that smaller log canonical threshold corresponds to worse singularities. You can compute via log resolution.
For example, take a smooth variety \(Y\) with a map \(\pi:Y\rightarrow \mathbb A^n_{\mathbb C}\) with \(\pi\) proper and birational, \(D = \mathbb V(f)\) with \(\pi^*D + K_{Y/\mathbb A^n}\) a simple normal crossings divisor. Equivalently, \(\big((f\circ \pi)\cdot \text{Jac}_{\mathbb C}(\pi)\big)\) is locally monomial on \(Y\).
We get a nice formula for the \(\text{lct}\), \[\text{lct}(f) = \min_{E_i}\left\{\frac{\text{ord}_{E_i}K_\pi + 1}{\text{ord}_{E_i}\pi^*D}\right\} \implies \text{lct}(f)\in \mathbb Q.\] E.g., if you take the cusp \(p = (0,0)\) in the cuspoidal curve \(D: x^3 = y^2\) in \(\mathbb A^2_{\mathbb C}\), you blow up once and get a line tangent to a conic, blow up again and you get two lines and a cubic all meeting at a point and finally blow up again and you get \(E_1, + E_2 + E_3 + \tilde{D}\), with \(E_i\) exceptional divisors copies of \(\mathbb P^1\) and \(D\) the strict transform of the original cubic. Letting \(\pi\) be the resolution of the cusp, we get \(\pi^*D = \tilde{D} + 2E_1 + 3E_2 + 6E_3\), \(K_\pi = E_1 + 2E_2 + 4E_3\) and \(\text{lct}(x^3 + y^2) = \min\left\{\frac{0 + 1}{1}, \frac{1 + 1}{2}, \frac{2 + 1}{3}, \frac{4 + 1}{6}\right\}.\)
Part 2: Prime Characteristic Fix a \(p > 0\), and take for instance \(f\in R = \mathbb F_p[x_1,...,x_n]\). The eth iterated Frobenius map \(F^e:R\to R\) is \(r\mapsto r^{p^e}\), which equivalently is a way to view \(R\subseteq R^{1/p} \subseteq R^{1/p^e} \subseteq ...\). We define \[R_{perf} = \projlim_{F}R = \bigcup_{e}R^{1/p^e},\] and we say that \(R\) is perfect if \(\text{Frob}\) is an isomorphism. Define then the “F pure threshold” \[\text{fpt}(f) = \sup_t\left\{t = \frac{a}{p^e} \in \mathbb Q_{> 0} ~ \middle | ~ f^a \not\in \big(x_1^{p^e}, ..., x_n^{p_e}\big)\right\}.\] This is equal to \(\sup_t\{t = a/p^e \in \mathbb Q_{>0} ~ | ~f^t \not\in (x_1,...,x_n) \cdot R_{perf}\}.\) E.g. \(f = x^m \in \mathbb F_p[x]\), \(x^{ma} \not\in \left(x^{p^e}\right) \iff ma < p^e,\) so \(\text{fpt}(x^m) = \frac1m\) and \(\text{fpt}(x_1^{m_1}\cdot...\cdot x_n^{m_n}) = \min\left\{\frac{1}{m_i}\right\}\).
Example: \(f = x^3 + y^2 \in \mathbb F_2[x,y]\), then \(f\in (x^2,y^2)\) but \(f\not\in (x^4, y^4 )\). This first case means that the \(\text{fpt}\) of \(f\) is less than \(1/2\), the second means it’s greater than \(1/4\).
A formula for \(f = x^3 + y^2\) in \(\mathbb F_p[x,y]\):
- it is \(1/2\) when \(p = 2\)
- it is \(2/3\) when \(p = 3\)
- it is \(5/6\) when \(p \equiv 1 \mod p\)
- it is \(5/6 - 1/6p\) when \(p \equiv 5 \mod p\).
Theorem
[Blickle, Mustata, Smith] \(\text{fpt}(f \mod p) \in \mathbb Q_{>0}\)
Theorem
[TW] If \(f\in \mathbb Z[x_1,...,x_n]\),
- Have \(\text{fpt}(f\mod p) \leq \text{lct}(f) \) for \(p \gg 0\)
- Have \(\lim_{p \to \infty} \text{fpt}(f\mod p) = \text{lct}(f)\).
Theorem
[Blickle-Schwede-Tucker] \(\text{fpt}\) can be described using regular alterations
Their theorem uses the following:
Theorem
[Hochster, Huneke] \(R = \mathbb F_p[x_1,...,x_n]\) with \[R^+ = \text{int closure of }R \text{ in } \Frac (R) = \bigcup_{R\subseteq S\subseteq \overline{\Frac(R)}, \text{ module finite}}S,\] then \(R^+\) is a perfect BCM algebra.
BCM algebra is a “big Cohen-Macaulay” module, an algebra over a Noetherian local ring where every system of parameters is a regular sequence.
Part 3: Mixed Characteristic
Theorem
[Bhatt] \(R = \mathbb Z_p [ [x_1,...,x_n] ]\), then \((R^+)^{\wedge p}\) is a perfectoid BCM algebra.
Definition
If \(f\in \mathbb Z_p[ [x_1,...,x_n] ]\) then \(\text{ppt}(f) = \sup\{t \in \mathbb Q_{> 0} | f^t \not\in (x_1,...,x_n)R^+\}\)
Theorem
Author + Schwede and the other dudes \(\text{fpt}(x^a + y^b) \leq \text{ppt} (x^a + y^b) \leq \text{lct}(x^a + y^b)\)